useoflogarithmsl00oberrich

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MACHINERY'S REFERENCE SERIES

E A C H N U M B E R I S A U N I T IN ' A S E R I F S O N E I . E C T R I C A I . A M )

S T E A M E N G I N E E R I N G D R A W I N G A N D M A C H I N E

D E S I G N A N D S H O P P R A C T I C E

NUMBER 53

USE OF LOGARITHMS ANDLOGARITHMIC TABLES

SECOND EDITION

C O N T E N T S

Th e U se of Lo ga ri th ms , by ERIK ORKRG 8

Tables of Logarithms 18

C o p y r i g h t , 19 12 T h e I n d u s t r i a l P r o a s . P u b l i s h e r s o f M u r H I N F j R Y,

1 9- 55 L a f a y e t t e S t r e e t . N e w Y o r k C i t y


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THE USB OF LOGARITHMS

It is not intended in the following pages to discuss the mathematical

principles cn which logarithms and expressions containing logarithms

are based, hut simply te impart a working knowledge of the use cf

logarithms, so that practical men, unfamiliar with this means foi

eliminating much of the work ordinarily required in long and cumber-

some calculations, may be able tc make advantageous use of the tables

of logarithms given in the latter part of the boik.

The object or' logarithms is to facilitate and snorten calculations

involving multiplication, division, the extraction cf roots, and the

obtaining of powers of numbers, as will be explained later; but odin

ary logarithms cannot be applied to operations Involving addition and

subt racti on. Before enteri ng directly upon the subject of the use of

logarithms in carrying out the various classes of calculations men

tioned. it will be necessary to deal with t he questi on of how to find

the logarithm of a given number frcm the tables; or, if the logarithm

is given, how to And the corresponding namber.

i A log arit hm consist s of two part s, a whole num ber and a decimal.

The whole number, which may be either a positive or a negative num

ber,* or zero, according to the rules which will be given in the follow

ing, is called the characteristic; the decimal is called the mantissa

The decimal or mantissa only is given in the tables of logarithms on

pages IS to 35, inclusive , and is alw ays positiv e. The log ari th m of 350,

for exampl e is 2.54407. Her e "2" is th e characteristics, and "54407".'isthe mantissa, this latter being found from the table on page 23

Rules for Finding' tte Characteristic

The characteristic is not given in the tables of logarithms, due

partly to the fact that it can so easily be determined without the aid

cf tables, and partly because the tables, sc to say, are universal when

the characteristic is left cut.

For 1 and oil numbers greater than 1 the characteristic is one less

than the number of placcs to the left of the decimal point in the given

num ber.

The characteristic of the logarithm of 237, therefore, is 2, because

2 is one less th an the num ber of figures in 237. The charac teris tic

cf the logarithm of 237.26 is also 2, because it, is only the number of

figures to the left of th e decimal po int tha t is considered.

The characteristic of the logarithm of 7 is 0, because 0 is one less

" S e e M A C H I N E R Y ' S l i e t e r e n e e S e r i e s N o . 5 4 , S o l u t i o n o f T r i a n g l e s , C h a p t e r

I I I , r o s i t i r e a n d N e g a t i v e Q u a n t i t i e s .

3 4 7 6 0 8


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METHODS, RULES AND EXAMPLES 5

for practical results i t is accurate enough to find the mantissa of thelogari thm of the first four f igures of the number, remembering, of

course, that i f the fif th f igure is more than 5, then the fou rth figure

should be increased by one unit .

To find the logarithm of a number from the tables, first locate the

three first f igures of the number for which the logari thm is required

in the l ef t han d column , and then find the fo urt h figure at the top of

the columns of the page. Then follow the column down from this last

f igure unti l opposite the three first f igu res In the left- hand colu mn.

The figure thus found in the body of the table is the mantissa of the

logari thm, the characterist ic having already been found by the rules

previously given.

If the number of which the logari thm is required does not contain

four figure s, annex cip hers to the rig ht so as to obta in four figu res.

If the manti ssa of the lo gari thm of 6 is required, fo r example, f ind

the manti ssa fo r 6,000. The man ti ss a is th e sam e for 6, 0.6, 0.06, 60,

600, 6,000, etc., as alre ady exp laine d. The differen ce in the loga rit hm

is taken care of by the change of characterist ic for these various

values. In order to save space in the tables, it will be seen by

referring TO them that the first two figures of the mantissa have been

given in the first col umns of log ari th ms only, th e 0-column. These

two figures should, however, always precede the three figures given

in each of the following columns.

A few examples will now make the use of the tables clearer

Example 1.Find the logarithm of 1852.

Following the rule already given, locate 185 in the left-hand column

of the table s fit will be foun d on page 19), and th en foll owi ng downwa rd the column head ed "2" at the top of the page

ma nti ssa opp osite 185. It will be seen that the ma nti ssa is .26764, *

the figures 26 being found in the column under "0" and prefixed to

the figures 764 found directly in the column under "2." The charac-

terist ic of the logari thm, according to the rules previously given, is 3.

Hence the logarithm of 1852, or, as it is commonly written, log 1852= 3.26764.

Example 2. -- Fi nd log :1.852.

As the figures in this number are the same as in that given in ex-

ample 1, the mantissa remains the same; but the characterist ic is 0.

Therefore , the requi red logar i thm, or log 1 .852=0.26764.

Example 3.Find log 93.14.

Loca te 931 in the left ha nd colu mn of the tabl es (page 3 4), and

th en followin g down wa rd the column hea ded "4" -at the top of the

page, find the requi red man tis sa opposite 931. It will be roun d th at

the ma nti ss a is .90914. The chara cte ris tic is 1. Henc e log 93.14

= 1.96914.Example 4.Find log 4.576.

Find as before the last three figures of the mantissa opposite 457

* A l l t h e m a n t i s s a s , o r t h e n u m b e r s t h e t a b l e s , a r e d e c i m a l s , a n d t h ed e c i m a l p o i n t h a s , t h e r e f o r e , b e e n o m i t t e d e n t i r e l y , s i n c e n o c o n f u s i o n c o u l d

a r i s e f r o m t h i s ; b u t i t s h o u l d a l w a y s b e p u t b e f o r e t h e f i g u r e s o f t h e m a n t i s s a

a s s o o n a s t a k e n f r o m t h e t a b l e . T h e p r a c t i c e o f e l i m i n a t i n g t h e d e c i m a l p o i n t

f r o m t h e t a b l e s i s c o m m o n t o a l l l o g a r i t h m i c t a b l e s .


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6 No. 53USE OF LOGARITHMS

in the lef t-ha nd column , and in the column u nd er "6" at the top of

the page. The figu res are *049. Th e sig n * ind ica tes that the two

figure s to be prefix ed are not 65, as wou ld ord ina ril y be th e case,

but 66, or the figures given in the next following line in the 0-column

This rul e sho uld alwa ys be bo rn e in min d. Hen ce, log 4.576 = 0.66049.

Example 5.Find log 72.

To find the mantissa, proceed as if it were to be found for 7200.

Thi s we find from the tables to be .85733. The cha rac ter ist ic of the

lo ga ri th m of 72 is 1. He nc e log 72 = 1.85733.

Example 6 Fi nd log 0.007631.

To find the mantissa, proceed as if it were to be found for 7631.

This we find from the tables to be .88258. The char act eri sti c is

according to the rule given for characterist ics of logari thms of num-

ber s less th an 1. Hen ce, log 0.007631 = 3.88258.

Example 7Find log 37,262.

While we wil l later explain how to find more exactly the mantissafor a nu mb er wit h five figures, at presen t we may cons ider it acc ura te

enough for our purpose to f ind the mantissa for four f igures, or for

3726. Th is is .57124. Th e cha rac ter ist ic of the log ari thm of 37,262 is

4. He nc e log 37,262 = 4.57124. Thi s, of cour se, is only an app ro xi ma -

tion. but is near enough for nearly al l shop and general engineer-

ing calculat ions.

If the given number had been 37,267 instead of 37,262, the logarithm

should have been found for 3727, as the fourth figure then should

have been increased by 1, when dropping the fifth figure, which is

larger than 5.

Below are given several examples of numbers with their logari thms.

A careful study of these examples, the student f inding the logari thms

for himself from the tables, and checking them with the results given,

wil l tend to make the methods employed clearer and fix them in the,

mi nd .

It should be understood that in logari thms of numbers less than 1,

the characteris t ic, only, is negative. The man tiss a is always posit ive,

so that 3.84510 actually means ( 3) + 0.84510.

Finding the Number whose Log is Given

When a logarithm is given, and it is required to find the corres-

ponding number, f irst f ind the first two figures of the mantissa in the

column headed "0" in the tables. Then find in the group of mantis sas,

al l havin g the same first two figures, the rem ain ing thr ee figures .

N u m b e r L o g a r i t h m

16.95 1.22917

2 0.30103

966.2 2.98507

151 2.17898

3.5671 0.55230

12.91 1.11093

3803.8 3.58024

0.007 -3.84510


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METHODS, RULES AND EXAMPLES 7

These may be in an y of the colum ns headed "0" to "9." Th e nu mb er

heading the column in which the last three figures of the mantissa

were found, is the last f igure in the number sought, and the number

in th e left -han d colu mn hea ded "N," in line wit h the figure s of th e

mantissa, gives the three first f igures in the number sought.

,When the actual f igures in the number sought have thus been deter-

mined, locate the decimal point according to the rules given for the

charac terist ic of logari thms. If the chara cteri st ic is grea ter than 3,

ciphers are added. For example, i f the figures correspond ing to a

certain mantissa are 3765, and the characterist ic is 5, then the num

ber sought must have 6 figures to the left of the decimal point, and

hence would be 376500. If the cha rac ter ist ic ha d been the n the

number sought, in this case, would have been 0.003765.

If tire mantissa is not exactly obtainable in the tables, find the near-

est man tiss a in the table to the one given and dete imin e the numbe r

corre spond ing to this. In most cases this gives ample accuracy. Amethod wil l be explained later whereby st i l l greater accuracy may be

obtained, but for the present i t wil l be assumed that the numbers

corresponding to the nearest mantissa in the tables are accurate,

enough for practical purposes.

A few examples will now be given in which it is required to find

the number when the logari thm is given.

Example 1.Find the number whose logarithm is 3.89382.

First find the fir^t two figures of the mantissa (89) in the column

headed "0" in the tables. Then find the remai nin g three figures (382)

in the manti ssa s which al l have 89 for the ir f irst two figures. The

figure s "382" are fou nd in the colu mn heade d "1," which 1hus is the

last figure in the number sought; the figures "382" are also opposite

the number 783 in the left-hand column, which gives the first three

figures in the numb er sought. The figures in this numbe r, thus, are

7831, and as the characteristic is 3, it indicates that there are. fourfigure s to the left of th e decimal poi nt, or, in other words , th at 7831 is

a whole nuj iber .

Example 2.Find the number whose logarithm is 2.75020.

First f ind the first two figures of the mantissa (75) in the column

head "0" in the tables. Then fina the remai nin g thre e figures (020)

in the mantissas, which al l have 75 for their f irst two figures. The

s in froi.t of the figure *020 in the line next aDove that in which 75-

was found indicates that these figures belong to the group preceded

by 75. The ref ore , ass020 is found in the column headed "6" and

opposite the number 562 in the left-hand column, the figures in the

nu mbe r req uir ed to be fou nd are 5626. As the cha rac ter ist ic is 2,

the decimal point is placed after the first three figures, and, hence,

the number whose logarithm is 2.75020 is 562.6.

Example 3.Find the numner whose logari thm is 2.45350.

After having located 15 in the column headed "0," it will be foundthat the last three figures (350) of the mantissa are not to be found

In th e table in the group prece ded by 45. The neare st value in the

table, which is 347. is, therefore, located, and the corresponding num-


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I


ber is found to be 284.1, the decin.al point being placed after the third

figure. because the chara cteri st ic of the logari th m is 2. Had the char-

acterist ic of the logari thm been 5 instead of 2, the number to be i- jnd

would have been 284,100.

Below are given a selection of exampl es of log ari thm s with tl ir

corres pondin g numb ers. The stud ent should find the numbe r? for

himself from the tables, and check them with the results given.

This will aia in fixing the rules and methods employed more firmly in

the mind.

L o g a r i t h mC o r r e s p ' H i d in g

L o g a r i t h m X u m o e r

1 . 4 3 2 0 1 2 7 . 0 4

4 . 8 9 1 7 0 7 7 . 9 3 0

2 - 7 6 0 5 7 0 . 0 5 7 6 2

0 . 1 2 0 9 6 1 . 3 2 1

2 . 9 9 0 9 9 9 7 9 . 5

T . 6 0 2 0 6 0 . 4

5 . 6 0 2 0 6 4 0 0 , 0 0 0

It being now assumed that the student has mastered the methods

for f inding the logari 'hms for given numbers, and the numbers for

given logari t hms from the tables, the use of loga ri th ms in mult ipl i-

cation and division will next be explained

Multiplication by Logarithm?

If two or more numbers are to be multiplied together, find the

logarithms of the numbers to be multiplied, and then add these

logarithms; the sum is the logarithm of the product, and, the, number'

corresponding to this logarithm is the required product.

Example 1 F in d th e pr od uc t of 2831 X 2.692 X 29.69 X 19-4.

This calculat ion is carried out by means of logari thms as follows:

log 2831. = 3.45194

log 2.692 = 0.43008

log 29.69 = 1.47261

log 19.4 =1 .2 87 80

6.64243

The sum of the logarithms. 6.64243, is the logarithm of the product,

and from the tables we then find that the product equals 4.390,000.

This result is, of course, only approximately correct at the last three

figures are added ciphers; but foi most engin eerin g calculat ions the

resu lt would give al l the accuracy requir ed. In most engineeri ng

calculat ions one or more factors are assumed f rom exper imenta l

values, and as these assumed values evidently must often vary be-

tween wide limits, it would show lack of judgment to require calcula-

t ions in which such assumed values enter, to be carried out with too

man y "significant ' ' f igures. Such values are fully as well expressed

in round numbers, with ciphers annexed to give the required valueto the figures found from the tables.

If one oi more of the characterist ics of the logari thms are negative,

these are subtracted instead of added to the sum of the character-


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METHODS, RUI.ES AND EXAMPLES 9

ist ics. The mantissas, as already mention ed, are always positive,

so that they are always added In the usual mann er. Tn order to

fully understand the adding of posit ive and negative numbers in the

following examples, the st uden t sho uld b(T fam ilia r with alet j lat jona

with positive and negative quantities, as explained in M*ACHI>~ERY'S.

Reference Series No. 54, Solution of Triangles, Chapter III .

Example 2.Find the product 371.2 X U.0972 X 8.

log 371.2 =2. 56 961

log 0.0972 = 2.98767 1,

log 3. =0 .4 771 2

2.03440

The nu mb er cor res pon din g to th e log ari thm 2.03440 is 108.2. Note

th at the first two figure s of the man tis sa of the log arit hm ar e 03.

Example 3 Fi nd t he pro du ct 12.76 X 0.012 X 0.6.log 12.76 =1.10585

l og 0 .01 2=2 . 0791 8 .1^ , . ' 6

log 0.6 =1. 77 81 5 t>

2.96318

The product, hence, is 0.09187.OJVT

Division Dy Logarithms

When dividing one number by another, the logari thm of the divisor

is subt racte d tro m the loga ri th m of the dividend. The rem ain der is

the logari thm of the quotient .

Fo r ex amp le, if we ar e to find th e quo tie nt of 7568 935.3, we first

find log 7568 an d the n sub tra ct fro m it log 935.3. The rem ai nd er

is then the logari thm of the quotient .

I t is advisable, however, to make a modification, as explained in

the following, of the logarithm of the divisor so as to permit of its

addit ion to, instead of i ts subtraction from the logari thm of the divi-

dend. Assume, for instance , th at an example, as below, were given:

375.2 X 97.2 X 0.0762 X 3

962.1 X 92 X 33.26

It would be perfectly correct to find the logarithms of all the fac-

tors in the numerator and add them together, and then the logari thms

of al l the factors in the denominator and add them together; and

finally subt ract th e sum of the loga ri th ms of the den omin ator fr om

the sum of the logari thms of the num era tor . The rema ind er is the

loga ri thm of the result of the calculat ion. This method, however,

involves two sep arat e addi t ion s and one subt racti on. I t is possible, by

a modification of the logari thms of the numbers in the denominator

to so arrange the calculat ion that a single addit ion wil l give the

logari tnm of the final result .In dealing with posit ive and negative numbers we learn that i f

we add a negative number to a posit ive number, the sum will be

the same as If we subtract the numerical value of the negative


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10 No. 53USE OF LOGARITHMSnum ber fro m the positiv e nu mb er ; t ha t is 5 -j- ( 2 ) = 5 2 = 3.

If ve reverse this proposit i on ve have 5. 2 = 5 4- ( 2 ) . If we

now assume that 5 is the logari thm of a certain number a and 2 the

logari thm of another number &, and if we insert these values in the

last expression, instead of 5 and 2, we have:

to g a log & = log a + ( log

Fru m this we see tha t instead of sub tra ctin g log h from log a we

can add the negat ive va lus of log h and obtain the sa me result . As

the mantissa always must remain posit ive, in order to permit direct

addit ion, the negative value of the logari thm cannot be obtained

by simp ly pla cin g a mi nu s sign befo re it. Ins tea d, it is obt ain ed in

the fol lowing manner :

If the characterist ic is posit ive, add 1 to i ts numerical value ar.d

place a minu s sign over it. To obta in the manti ssa , sub tra ct the

given mantissa from 1.00000.

Example 1.This log ar it hm of 950 = 2.97772. Fi nd ( log 950).

According to the rule given, the characte rist i c wil l be 3. The man-

tis sa will be 1.00000 .97772 = .02228. Th e last calc ula tio n ean be

carrie d out mentally witho ut wr it in g i t down at al l , by simply finding

the figure which, added to the last f igure in the given mantissa would

make the sum 10, and the figures which added to each of the other

f igures in the mantis sa, would make the sum 9. as shown below:

9 7 7 7 20 2 2 2 8

9 9 9 9 10

As this calculat ion is easi ly carried out mentally, the method

described, when fully mastered, greatly simplifies the work where

operations of bo fh mult ipl icat ion and division are to be performed in

the same example. j \Example 2.The log ari thm of 2 is 0.30103. Fi nd ( log 2).

According to the given rules the character i s t ic i s I , a n d the

mantissa, .69897.

The following examples should be studied unti l thoroughly under-

s t ood. U O

log 270. =2 .4 31 36 log 270. =3 .5 68 64

log 10. =1 .0 00 00 leg 10. =T. OOOO O^^ ,

log 26.99 =1 .4 31 20 log 26.99 =5 .5 68 80

In the example in the second iine an exception from the rule for

obta inin g the man tiss a of the negati ve loga ri th m is made. I t is

obvi ous, howe ve r, th at if log 10 = 1.00000, th en ( log 101 =

1.00000. In th e exa mpl e in the last line ther e is an oth er deviat ion

fro m the l i teral under sta ndi ng of the rule for the manti ssa. As the

last figure in the positive lo ga rit hm is 0, the last figure in ( log

26.99) is also 0, and the next last figure is treated as if it were thelast , making the next lasv figure in the negative logari thm 8

If the characterist ic of the logari thm!s negative, subtract 1 from

its numeric al value, and mak e i t posit ive. The manti ssa is obtained

by the same rule as before.


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ME T H O D S , RUI.ES AND EXAMPLES 11

Example 1 Th e log ari thm of 0.003 3.47712. Fi nd ( -- log 0.003).

Accordin g to the rule iust given the chara cteris t ic wil l be 2. Th e

ma nt is sa -will be .52288. Hen ce ( log 0.003) = 2.52288.

The following examples should be studied unti l ful ly understood:

log 0.3 = T.47712 l og 0.3 =0 .5 22 88

log 0.0006963= 3.84280 lo g 0.0006963 =3. 157 20

log 0.6607 =1. 82 00 0 l og 0.6607 =0. 18 000

"When sufficient practice has been obtained, the negative valae of

a logarithm can be read off almost as quickly from the tables as the

positive value given, and the subsequent gain of time, and the ease of

the calculat ions following, more than just ify this short-cut method

Examples? of the Use of Lcg-arit'nms

We will now giv e a nu mb er of exa mple s cf the use of log ari thm s in

calculat io ns involving mult ip l icat i on and division. Xo comme nts

will be made, as it is assumed that the student has now grasped the principles sufficiently to be able to follow the methods used without

fur ther explanat ion.

Example 1.

0.0272 X 27.1 X 12.6

2.371 X C.007

log 0.0272 = 2-43457

log 27.1 = 1 43297

log 12.6 =1. 100 37

Oog 2.371 =1 .6 25 07

log 0.007 =2 .1 54 90

2.74788The result, then, is 559.6.

Example 2.

0.3752 X 0.063 x 0.012

0.092 x 1289

le g 0.3752 = 1:57426

log 0.063 =2 .79 93 4

log 0.012 =2 .0 79 18

log C.092 = 1.03621

log 12S9.0 =5 .8 89 75

6.37874The result, then, is 0.000002392.

Example 3.

3.463 X 1.056 X 14.7 x" 144 X 10

log 3.463 = 0.53945

log 1.056 = 0.02366

log 14.7 = 1.16732

log 144.0 =2. 158 36log 10.0 =1.00000

The result, then, is 77,410.4.88879


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Example 4.

0.00005427 X 392 X 2.5 X 200 X 200

log 0.00005427 = 3.73456

log 392. =2. 593 29

log 2.5 =0. 39 794

log 200. = 2.30103

log 200. = 2.30103

3.32785Hence, the result is 2127.

Obtaining: -he Pow er? of Numbe rs

Expressions of the form 6.513 can easily be calculated by means of

log ari thm s. Th e xinal- (') is called exponent..* In th is case the

"third power" of 6.51 is required.

A namber may be raised to any power by simply mult iplying the

loga ri th m oi the num ber hy the exponent of the number . The product

gives the logari thm of the value of the power.

Example 1.Find the value of 6.51\

log 6.51=0.81358

3 X 0.81358 = 2.44074

The log ari thm 2.44074 is the n th e log ari thm of 8.51*. Hen ce 6.51'

equals the number corresponding to this logari thm, as found from the

ta bl es , or 6.51* = 275.9. "

Example 2.Find the value uf 121-2".

lo g 12 = 1.07918

1.29 X 1.07918 = 1.39214

Hence, 12 ' - ' = 24.67.

The multiplication. 1.29 X 1.07918 is carried out in the usual arith-

meti cal way. The exam ple abov e is one of a type which canno t be

solved by any means except by the use of logar i thms . An expression

of the form 6.51* can be found by arithmetic by multiplying 6.31

X 6.51 X 6.51, but an expression of the torm 12 1-2" does not permit

of being calculated by any ari t hmet ical meth od. Log ari thm s are here

absolutely essential .

One difficulty is met with when raising a number less than 1 to a

given power. The loga ri thm is then composed of a negativ e term , the

charac terist i c, and a posit ive term, the mant issa . For examp le: Fin d

th e value 0.31". Th e loga ri th m of 0.31 = T. 49136. In th is case, mul ti-

ply, separately, the characterist ic and the mantissa by the exponent,

as shown below. Then add the produc ts.

log 0.31=1.49136

Multiplying characterist ic and mantissa separately by 5 we have:

5 X 1 = 3

5 X .49136 = 2.45680

lo g 0.31= = "$.45680Hen ce, 0.31' = 0.002863.

r S e e M A C HI N E R I E S R e f e r e n c e S e r i e s N o . 5 2 , . A d v a n c e d S h o p A r i t h m e t i c t o r

t h e M a c h i n i s t . C h a p t e r I I I .


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Tf the exponent is not a whole number, the procedure will be some-

what more complicated. The principle of the method, however, re-

mains the same.

Example: Find the value of 0.065-

31

log 0.06 = 2,77815

T h e n L j t1

2.31 X 2 = 2.31 X ( 2) = - f 4 . 6 2

2.31 X 0.77815 = 1.79753

In thi s case, the first produ ct, 4.62, is neg ati ve both as re ga rd s the

whole num ber and the decimal. In orde r to mak e the decimal posit ive

so that we may be able to add it directly to the secoml product, 1.79753,

we must, use the same rule as given for changing a logarithm with a

posit ive chara cteris t ic to a negat ive value. Hence 4 .6 2= 3- 38 . We

can now add the produc ts:

5.381.79753

log 0.06?-

31Jfj.17753

Hen ce 0.0 6^" = 0.001503.

As a further example, find 0.07 s-51 .

lo g 0.07 = 2.84510

Then

3.51 x 2 = 3 .51 X ( 2 ) = 7.0 2=3 .98

3.51 X .S1510 =2 .9 66 30

log 0.07s-

61=: 5.91630

Hence 0.073-51 = 0.00008837.

Extracting Boots by Logarithm*

Roots of nu mb ers , as for exa mple 37, can easil y be extr acte d by

mea ns of loga ri thm s. The small (3) in the radi cal (V ) of the root-

sign is called the index of the root. In the case of the squ are root

the index is (2), but it is not usually indicated, the square root being

merely expressed by the sign V.

Any root of a number may be found by dividing its logarithm by

the index of the root; the quotient is the logari thm of the root .

Example 1.Find f/~276.

log 276 =2. 440 91

2.44091 -f -3 = 0.81364

Henc e log -^ Is Ss and ^'"2 76 = 6.511.

Example 2.Find f ' 0.67

log 0.67 = 1.82607

In this case we cannot divide directly, because we have a negative

characterist ic and a posit ive mantissa. We then proceed as follows: Add

numerically as many negative units or parts of units to the character-istic as is necessary to make it evenly contain the index of the root.

Then add the same number of positive units or parts of units to the

manti ssa. Divide each separ ately by the index. The quotie nts give


13/39


the characterist ic and mantissa, respectively, of the logari thm of the

root.

Proceeding with the example above according to this rule, we have:

T + 5 = 3 ; 3 4 - 3 = 1 .

.$2607 + 2 = 2.82607 ; 2.82607 -r 3 = .94202.

He nc e, log f "0X7 = 1.94202, an d f 0.67 = 0.875.it-

Example 3. Fin d j 0.2.

log 0.2 = 1.30103.

If we add ( 0.7) to the char act eri sti c of the loga rit hm roun d, it

will be evenly divisible by the index of the root.

H ence :T -f ( 0.7) = 1 .7 ; 1. 7-4 -1. 7 = 1.

.30103 + 0.7 = 1.0010 3; 1.00103 jr 1.7 = .58884.

He nc e, log V 1)72 = T.585S4, an d ) 0.2 = 0.3S8.A number of examples of the use of logarithms in the solution cf

everyday problems in mechanics, are given in MACHINERY'S Reference

Series No. 19, Use of Formulas in Mechanics, Chapter II, 2nd edition.

When exponents or indices are given in common fractions, i t is

usually best to change them to decimal fractions before proceeding

further with the problem.

Interpolation

If the number for which the logari thm is required consists of f ive

figu res, it is possible, bj me an s of the sma ll tables in the right -ha nd

column of the logari t hm tables, heade d "P. P." (prop ort ion al part s) ,

to obtain the logari thm more accurately than by taking the nearest

value for tour f igures, as has previously been done in the examples

given. The method by which the loga ri th m is then obtained is cal led

interpola t ion.

In the same way, if a logarithm is given, the exact value cf whichcannot be found in the tables, the number corresponding to the logar-

i thm can be found to f ive figures by interpolat ion, al thougn the main

tables cont ain only num ber s of four f igures.

The lo ga rit hm of 2853 is 3.45530, an d the lo ga rit hm of .2854 is

3.45545, as fou nd fro m the tables. Ass ume that the logar ith m of 2853.6

were xequired. I t is evident that the loga ri thm of this lat te r num ber

ma st have a val ue bet wee n th e log ari thm s of 2853 and 2854. It mu st

be somewhat great er than the logari t hm of the forme r numb er, and

somew hat small er than that of the lat ter . Whil e the loga ri thm s, in

fe ne ra l are not propo rt iona l to the num ber s to which they corres-

pond, the difference is very slight in cases where the increase in the

numbers is small; so that, in the case of an increase from 2853 to 2854,

the logarithms for the decimals 2853.1, 2853.2, etc., may be considered

proport ion al to the numbe rs. I t is on this basis tha t the small tables in

the right-hand column headed "P.P." are calculated, and the logari thmof 2853.6, for example, is found as follows:

Find first the difference between the nearest larger and the nearest

ent aile r lo ga ri th ms . Log 2854 = 3.45545 an d log 2853 = 3.45530. The


14/39


differe nce is 0.00015. The n in the small table head ed "15" in the right -han d colum n find the figure opp osite 6 (ti being the last or fifth figure

in the given nu mb er ). Thi s figure is 9.0. Add thi s to the man tis sa

of the smaller of the two logari thms already found, disregarding the

decimal point in the mantissa, and considering i t , for the while being

as a whole nu mbe r. The n 45530 + 9.0 = 45539. Thi s is the mant is sa

of the logarithm of 2853.6, and the complete logarithm is 3.45539.

Example.Find log 236.24.

Lo g 236.2 = 2.3732 8; log 236.3 = 2.37346; di ff er en ce = O.00018.

In table "18" the pr opo rti ona l part oppos ite 4 is 7.2. The n 37328 -f 7.2

= 37335.2. The decima l 2 is not used , but is dropped. Hen ce log

236.24 = 2.37335.

If the proport ional part to be added has a decimal larger than 5, i t

should not be dropped before the figure preceding ft has been raised

one uni t. Fo r exam ple, if th e log ari thm of 236.26 had been req uir ed,

then the proport ional part would have been 10.8 and the mantissa

sought 37328 + 10.8 37338.8. Now the decimal 8 cannot be dropped

before the figure 8 pre ced ing it has been rais ed to 9. The n log

236.26 = 2.37339.

If the number for which the lorgari thm is to be found consists of

more than five figures, f ind the mantissa for the nearest number of

f ive figures but choose the ch arac teri st ic ac cordin g to the total num-

ber of figures to the left of the deci mal point. Fo r exam ple, if the

logari thm of 626.923 is required, f ind the mantissa, by interpolat ion,

for 62692 If th e lo ga rit hm for 626,928 is req uir ed, fi nd th e man tis sa

for 62603, always remembering to raise the value of the last figure,

if the figure drop ped is more than 5. The char act eri sti c in each of

these examples would, of course, be 5, as it is chosen according to the

total number of figures to the left of the decimal point in the given

numbers, which is 6.

To find a number whose logari thm is given more accurately lhan to

t 'our f igures, when the given mantissa cannot be found exactly in th"

tables, f ind the mantissa which is nearest to. but less than the given

mant i ssa . Subt ract th i s manl i ssa f iom the neares t la rger mant i ssa

in the tables and find in the right-hand column the small table headed

by this difference. Then subtrac t the near est smalle r manti ssa from

the given logari thm, and find the difference, exact or approximate, in

the " propo rt ion al pa rt" tab le ( in the right han d c.olamn of this

tab le). The correspo nding figure in the left-hand column of the "pro-

port ional part" table is the fif th f igure in the number sought, the other

four f igures being vhose corresponding to the logari thm next mailer

to the given logari thm.

Example. Find the number whose logarithm is 4.46262.

The mantissa can not be found exactly in the tables; therefore, fol-

lowing the rules just given, we see that the nearest smaller mantissa

in th e tabl es equa ls 46255. Th e next large r is 46270. The difference

between the m is 15. The diffe renc e betw een the ma nt iss a of the givenloga rith m. 46262 and the next small er man tis sa, 46255 is 7. Now, in

the proport ional parts table opposite 7.5 in the right-hand column of


15/39

16 No. 5 3 U S E OF LOGARITHMS

the table headed 15. we tlnd that the fifth figure of the number sought

would be 5. Th e fou r first figure s ar e 2901. Hen ce the nu mb er soug ht

is 29,015.

The following examples, i f caref ully studied wil l give the studen t

a clear conception of the method of interpolat ion.

N u m b e r L o g a r i t h m

52,163 4.71736

26.933 1.42996

0.012635 2.10157

12.375 1.09254

6.9592 0.84256

The student should find for himself f irst the logari thms correspond-

ing to the given numbers, and then the numbers corresponding to the

given logari th ms. In this way a check on the accuracy of the work

can be obtained by comparing with the results given.

General Bemarks

In the system of logarithms tabulated on pages 18 to 35, the base-

of the logari thms is 10; that is, the logari thm is actually the exponent

which would be affixed to 10 in order to give the number correspond

ir.g to the loga rit hm. Fo r exa mpl e log 20 = 1.30103, whi ch is the

same as to say that lO1-301"3 = 20. Log 100 = 2, and, of course , we

know that lp2

= 100. As 101

= 10, the loga rit hm of 10 1. The

loga ri th m of 1 = 0. The system of log ari t hms havin g 10 tor i ts base

is cal led the Briggs or the common system of logari thms.

While the accompanying logari thm tables are given to f ive decimals,

it should be understood that the logarithm of a number can be calcu-

lated with any degree of accuracy, so that large logari thm tables give

the logarithm with as many as seven decimal places, an-1 some, used

for very accurate scientif ic investigations, give as many as ten deci-

mals. I t wil l be noticed that in the accompan ying tables the figure5, when in the fifth decimal place, Is eit her wr itt en 5 or 3. If th e

six th place is 5 or more, the nex t larg er numb er is used in t he fifth

place, an d the logar ith m is the n wr itt en in the form 3.90853. The

dash over the 5 shows that the logari thm is less tha n given. If the

sixth figure is less than 5, the logarithm is written 3.91025, the dot

over the 5 showing that the logari th m is more than given. In calcu-

lat ions of the type previously explained, this, however, need nor be

taken into consideration and these signs should be disregarded by the

s tudent .Hyperbolic Logarithms

In certain mechanical calculat ions, notably those involving the calcu-

lat ion of the mean effective pressure of steam in engine cylinders, use

is made of logarithms having for their base the number 2.7183, com-

monly d esigna ted e, ar .d found by abst ract math ema tica l an? lysis.

These logari thms are termed hyperbolic, Xapierian or natural; t i preferable name, and that most commonly in use in the United States

is hyperbolic logari thms. The hyperbolic logari thms are usually desig-

nat ed "hy p. log." Thus , whe n log 12 is req uire d, it alw ays refe rs to


16/39


common logari th ms, hut when the hyp. log 12 is required , referenceis made to hyperbo lic logar i thms . Sometim es, the hyperbolic loga-

rithm is also designated "log," and "rial, log."

To convert the common log ari t hms to hyper bolic logari thms, th e

fo rm er shou ld be mul tip lie d by 2.30258. To conve rt hyperbol ic loga-

ri t hms to common logari th ms, mult i ply by 0.43429. These mult i pl ier s

will be found of value in cases where hyperbolic, logarithms are re-

quired in form ulas. Hyperb olic loga ri th ms find extensive use in

higher mathemat ics .

SECTION II

TABLES

COMMON LOGARITHMS

1 TO 10,000


17/39

18 No. USE OF LOGARITHMS

N L 0

1 0 0 o o o o o 3 4 3 I 3 : 7 3

1 ' 4 3 2 4 7 f 8 S 6 1 6 0 4

102 800 903 94=; ^33 ^030103 01 284 326 3b8 io 452104 703 74S 787 828 870

1 0 5

106 V108109

0 2 I I 9 1 6 0 2 3 2 2 4 3 2 3 4

531

572 612 633 C949 3 8 0 7 9 # 0 1 9 TOOO * 1JO

34 333 423 403 503743 782 822 802 902

110 O4I39 I79 218 258 2Q7111 532 57I 6lO 6?0 689112 922 961 999 #038 #077113 05308 340 38; 423 4*31114 690 729 767 805 843

115 06 OTO jzn 145 183 221110 446 483 521 558 59Sii 7 819 856 893 v3c 967lib 07,188 225 2->2 2-1I 335" 9 55? J9i ^28 604 700

120 17?127 3 8 - 4 i f 4 4 9 1 * 3 S 712b 721 75 789 823 857129 11050 cj? 126 10 J 193

130 394 428 461 4v4 528131 727 760 793 82b 3 00132 12057 9 T-y 156 189

133 384 413 4JJ 483 516134 710 743 775 808 840

135 13 033 060 098 130 1621 3 ^ 3 5 4 3 8 b 4 1 8 450 4 3 1137 672 704 73 J 767 799138 988 s,oi9 ^051 sO82 ii4139 14 301 333 304 39j 420

140 613 644 675 706 73-71 4 1 9 2 2 9 5 3 9 8 3 D I 4 , 0 4 5

142 15 22 9 25 9 2yO 32 0 3CI

143 534 ^ 4 5 9 4 6 2 5 65;144 830 36-5 897 927 957

145 16 137 167 197 227 256146 43? 46=: 49? 524 554147 732 761 791 320 850148 17 C20 05S 38? 114 14;149 319 343 37-7 406 43S

150

N

609 638 6(r 606 725

I 0

6 8 9

2 1 7 2 T O 3 0 3 3 4 6 3 8 9

647 089 7., 775 SI7


18/39

LOGARITHMIC TABLES 19

L. 0 1 2 3 4 5 6 7 8 9 P P

15015 1

15 3

15 4

17 609 638 667 696 7258 9 8 9 2 6 9 5 5 9 8 4 , 0 1 3

18 I&4 21 3 24 1 27C 2< 8

4 6 9 4 g 8 5 2 6 5 5 4 5 8 3

752 780 608 837 863

75 4 7 82 8 u 840 869

,0 41 07r> ,0 99 , , 127 *I 56

327 355 384 412 441

61 j 6 3 9 6 6 7 6 9 b 7 2 4

S93 921 949 977 a o o 5

29 28

1 2 , 9 2 8

2 5,8 5.63 8,7 8 ,4

4 1 1 , 6 1 1 , 2

5 14,5 14, 0

6 17 4 lb,87 213,3 19 ,6

8 23, 2 2u,4

9 2 6 I 2 5 , 2

27 26

1 2 , 7 2 , 6

2 5,+ 52

3 8 /1

7 8

1 10,8 10,4

5 13 ,5 I3.6,2 15,6

j 18 21 6 20 8

9 24, 3 23,4

25

1 2 52 5 , 0

3 7 5

41 0

5 1^,56 15,0

7 F 7 58 2 0 , 0

0 2 2 , 5

24 23

1 2 , 4 2 , 3

2 4 , 8 4 , 63 7 . 2 6 . 9

4 9, 6 9 2

5 1 2 , 0 l i e6 14,4 13,8

7 l b 8 16 *

8 1 9 J 18 49 21 '> 20 ,/

22 21

1 2.2 2,1

2 4 , 4 4 , 2

3 6,6 6 , 3, 8 , 8 84.

5 11 u 10 3

6 13,2 12,6

r5, 4 M 7

1 7 , 6 i b , S

9 19, 8 i3 9

1 55

1 50

1 57

15 8

15 9

19 C33 0 61 08 9 1 17 145;

j I 2 34O 3O8 39b 42 4

5 9 6 1 8 6 4 $ 6 7 3 7 0 0

866 893 921 948 976

20 140 167 194 222 249

173 201 229 257 28 ;

45J 479 507 535 56 -

7 2 8 7 5 b 7 83 8 i ' 8 3 8

* C 3 * o 3 o , 0 5 8 , 0 85 i l l 2

27b 303 330 358 383

29 28

1 2 , 9 2 8

2 5,8 5.63 8,7 8 ,4

4 1 1 , 6 1 1 , 2

5 14,5 14, 0

6 17 4 lb,87 213,3 19 ,6

8 23, 2 2u,4

9 2 6 I 2 5 , 2

27 26

1 2 , 7 2 , 6

2 5,+ 52

3 8 /1

7 8

1 10,8 10,4

5 13 ,5 I3.6,2 15,6

j 18 21 6 20 8

9 24, 3 23,4

25

1 2 52 5 , 0

3 7 5

41 0

5 1^,56 15,0

7 F 7 58 2 0 , 0

0 2 2 , 5

24 23

1 2 , 4 2 , 3

2 4 , 8 4 , 63 7 . 2 6 . 9

4 9, 6 9 2

5 1 2 , 0 l i e6 14,4 13,8

7 l b 8 16 *

8 1 9 J 18 49 21 '> 20 ,/

22 21

1 2.2 2,1

2 4 , 4 4 , 2

3 6,6 6 , 3, 8 , 8 84.

5 11 u 10 3

6 13,2 12,6

r5, 4 M 7

1 7 , 6 i b , S

9 19, 8 i3 9

16016 1

16 2

16 3

1 04

16 516 6

165

16 S

1 69

412 439 466 493 5-2

0?

3 7 i o 7 3 7 7 0 3 7 9 c

952' 978 -00 5 3,032 , , 0 5 9

21 219 24 3 2 7 2 2Q9 326;

48 4 511 >37 504 590

=48 573 602 629 656

817 84 4 871 898 923

o 8 J * H 2 . , 139 , , 165 * I 9 2

352 378 405 431 458

6 1 7 6 4 3 6 0 9 6 9 6 7 2 2

29 28

1 2 , 9 2 8

2 5,8 5.63 8,7 8 ,4

4 1 1 , 6 1 1 , 2

5 14,5 14, 0

6 17 4 lb,87 213,3 19 ,6

8 23, 2 2u,4

9 2 6 I 2 5 , 2

27 26

1 2 , 7 2 , 6

2 5,+ 52

3 8 /1

7 8

1 10,8 10,4

5 13 ,5 I3.6,2 15,6

j 18 21 6 20 8

9 24, 3 23,4

25

1 2 52 5 , 0

3 7 5

41 0

5 1^,56 15,0

7 F 7 58 2 0 , 0

0 2 2 , 5

24 23

1 2 , 4 2 , 3

2 4 , 8 4 , 63 7 . 2 6 . 9

4 9, 6 9 2

5 1 2 , 0 l i e6 14,4 13,8

7 l b 8 16 *

8 1 9 J 18 49 21 '> 20 ,/

22 21

1 2.2 2,1

2 4 , 4 4 , 2

3 6,6 6 , 3, 8 , 8 84.

5 11 u 10 3

6 13,2 12,6

r5, 4 M 7

1 7 , 6 i b , S

9 19, 8 i3 9

16016 1

16 2

16 3

1 04

16 516 6

165

16 S

1 69

748 775 80: 827 85422 e n 0 37 C63 0 89 11J

272 298 324 350 -7b

S3I 557 583 608 634

789 814 340 86b 891

880 yo6 932 958 985I4 I 16 7 IC14 22C 24 6

4 0 1 4 2 7 4 5 3 4 7 9 5 0 5

6 6 0 6 8 6 712 7 3 7 7 6 3917 943 9 68 0 94 *

r9

29 28

1 2 , 9 2 8

2 5,8 5.63 8,7 8 ,4

4 1 1 , 6 1 1 , 2

5 14,5 14, 0

6 17 4 lb,87 213,3 19 ,6

8 23, 2 2u,4

9 2 6 I 2 5 , 2

27 26

1 2 , 7 2 , 6

2 5,+ 52

3 8 /1

7 8

1 10,8 10,4

5 13 ,5 I3.6,2 15,6

j 18 21 6 20 8

9 24, 3 23,4

25

1 2 52 5 , 0

3 7 5

41 0

5 1^,56 15,0

7 F 7 58 2 0 , 0

0 2 2 , 5

24 23

1 2 , 4 2 , 3

2 4 , 8 4 , 63 7 . 2 6 . 9

4 9, 6 9 2

5 1 2 , 0 l i e6 14,4 13,8

7 l b 8 16 *

8 1 9 J 18 49 21 '> 20 ,/

22 21

1 2.2 2,1

2 4 , 4 4 , 2

3 6,6 6 , 3, 8 , 8 84.

5 11 u 10 3

6 13,2 12,6

r5, 4 M 7

1 7 , 6 i b , S

9 19, 8 i3 9

17017 1

17 2

17 3

17 4

23 C43 07 0 096 121 147

3 0 c 325 350 3 7 6 ,+Ci

553 578 bo3 629 6 5 4

80 3 3C 855 880 905.

2 4 0 5 J c 8 o 103 130 15;

172 Iy8 223 249 274

4 2 6 4 5 2 4 } 5 0 2 5 2 8

67 9 704 72 9 ',-'54 77 4

4 3 9 5 5 9 8 0 * o o 5 , 0 3 0

1S0 204 229 254 J7v

29 28

1 2 , 9 2 8

2 5,8 5.63 8,7 8 ,4

4 1 1 , 6 1 1 , 2

5 14,5 14, 0

6 17 4 lb,87 213,3 19 ,6

8 23, 2 2u,4

9 2 6 I 2 5 , 2

27 26

1 2 , 7 2 , 6

2 5,+ 52

3 8 /1

7 8

1 10,8 10,4

5 13 ,5 I3.6,2 15,6

j 18 21 6 20 8

9 24, 3 23,4

25

1 2 52 5 , 0

3 7 5

41 0

5 1^,56 15,0

7 F 7 58 2 0 , 0

0 2 2 , 5

24 23

1 2 , 4 2 , 3

2 4 , 8 4 , 63 7 . 2 6 . 9

4 9, 6 9 2

5 1 2 , 0 l i e6 14,4 13,8

7 l b 8 16 *

8 1 9 J 18 49 21 '> 20 ,/

22 21

1 2.2 2,1

2 4 , 4 4 , 2

3 6,6 6 , 3, 8 , 8 84.

5 11 u 10 3

6 13,2 12,6

r5, 4 M 7

1 7 , 6 i b , S

9 19, 8 i3 9

17 517 b

17:17 S

17 9

304 329 353 378 4C35 5 1 5 7 6 bo i 6 2 5 t j c

7 97 822 846 S71 R9J

25 "U2 0 66 09 I 11=, 13 9265 3 1 0 3 3 4 3 5 8 3 8 2

428 452 477 302 527

674 bgg 72 4 7 48 77 J

9 2 0 9 4 4 9 6 9 9 9 3 i 8

164 188 212 237 261

4 0 6 4 3 1 4 5 s 4 7 4 5 0 3

29 28

1 2 , 9 2 8

2 5,8 5.63 8,7 8 ,4

4 1 1 , 6 1 1 , 2

5 14,5 14, 0

6 17 4 lb,87 213,3 19 ,6

8 23, 2 2u,4

9 2 6 I 2 5 , 2

27 26

1 2 , 7 2 , 6

2 5,+ 52

3 8 /1

7 8

1 10,8 10,4

5 13 ,5 I3.6,2 15,6

j 18 21 6 20 8

9 24, 3 23,4

25

1 2 52 5 , 0

3 7 5

41 0

5 1^,56 15,0

7 F 7 58 2 0 , 0

0 2 2 , 5

24 23

1 2 , 4 2 , 3

2 4 , 8 4 , 63 7 . 2 6 . 9

4 9, 6 9 2

5 1 2 , 0 l i e6 14,4 13,8

7 l b 8 16 *

8 1 9 J 18 49 21 '> 20 ,/

22 21

1 2.2 2,1

2 4 , 4 4 , 2

3 6,6 6 , 3, 8 , 8 84.

5 11 u 10 3

6 13,2 12,6

r5, 4 M 7

1 7 , 6 i b , S

9 19, 8 i3 9

18018 1

18 2

18 4

52

7 5 51

5 7 5 6 0 0 6 2 4

768 792 816 &40 864

26 007 031 053 C79 102

245 2tx, 293 316 34-t

4 8 2 5 0 3 5 2 9 5 5 3 5 7 6

6 4 8 6 7 2 6 9 6 7 2 0 7 4 4

8 8 8 w i 2 9 3 3 9 5 9 9 8 3

12 6 I^O I74 198 221

364 387 4? I 4 3 5 4 5 8

6 0 0 6 2 3 6 4 7 6 7 0 6 9 4

29 28

1 2 , 9 2 8

2 5,8 5.63 8,7 8 ,4

4 1 1 , 6 1 1 , 2

5 14,5 14, 0

6 17 4 lb,87 213,3 19 ,6

8 23, 2 2u,4

9 2 6 I 2 5 , 2

27 26

1 2 , 7 2 , 6

2 5,+ 52

3 8 /1

7 8

1 10,8 10,4

5 13 ,5 I3.6,2 15,6

j 18 21 6 20 8

9 24, 3 23,4

25

1 2 52 5 , 0

3 7 5

41 0

5 1^,56 15,0

7 F 7 58 2 0 , 0

0 2 2 , 5

24 23

1 2 , 4 2 , 3

2 4 , 8 4 , 63 7 . 2 6 . 9

4 9, 6 9 2

5 1 2 , 0 l i e6 14,4 13,8

7 l b 8 16 *

8 1 9 J 18 49 21 '> 20 ,/

22 21

1 2.2 2,1

2 4 , 4 4 , 2

3 6,6 6 , 3, 8 , 8 84.

5 11 u 10 3

6 13,2 12,6

r5, 4 M 7

1 7 , 6 i b , S

9 19, 8 i3 9

18 51 86

18 7

1B8

18 9

717 74 i 7&t 788 81 1

9 5 1 9 7 5 9 9 8 * o 2 i , 0 4 3

27 1&4 207 231 2^4 27 7

41 6 43 9 462 485 =;o8

6 4 6 66 9 6 9 2 7 1 5 7 3 8 ,

834 858 881 905 928

o b 8 * c x i 1 * I I 4 , , 138 # I 6 X

3 0 c 3 2 3 3 4 0 3 7 0 3 9 3

531 55 1 577 6 2 37 6 1 7 8 4 8 0 7 8 3 c 8 5 2

29 28

1 2 , 9 2 8

2 5,8 5.63 8,7 8 ,4

4 1 1 , 6 1 1 , 2

5 14,5 14, 0

6 17 4 lb,87 213,3 19 ,6

8 23, 2 2u,4

9 2 6 I 2 5 , 2

27 26

1 2 , 7 2 , 6

2 5,+ 52

3 8 /1

7 8

1 10,8 10,4

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6 1 0 , 2

7 1 1 9

8 15 ,6

9 1 5 3

161 1 ,6

2 . 2

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6 9 ,6

7 XI 2

1^,3

9 1 4 , 4

15

1 1 5

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4

5 7 5

9 , o

7 i e / S

8 12 ,0

9 13 ,5

14

1 i , 4

2 2 ,8

3 4^2

-1 5, 6

5 : P6 0,4

7 v .*

8 11 ,2

9 12 6

28028 1

2 8 2

28 3

2 8 4

71 6 731 747 76 2 778

871 88 6 90 2 917 932

4 5 0 2 5 0 4 0 0 5 6 171 0 8 6x

9 4a 2

5 240

33 2 347 362 378 393

793 8c 9 82 4 S40 855

. 4 8 9 6 3 9 7 9 9 9 4 * '

102 117 133 148 163

25 5 27-. 286 301 31;408 423 439 454 469

18

1 J,j82

3 ,6

3 5 /4

4 7 , 2

5 9 , o

6 n i,8

7 1 2 , 68 14 ,4

9 16 ,2

17

1 1,7

2 3 , 4

3 5 ,1

4 6 ,85 8 ,5

6 1 0 , 2

7 1 1 9

8 15 ,6

9 1 5 3

161 1 ,6

2 . 2

3 4 ,8

4 6 , 45 3 ,o

6 9 ,6

7 XI 2

1^,3

9 1 4 , 4

15

1 1 5

2 3 ,

3 4 5

4

5 7 5

9 , o

7 i e / S

8 12 ,0

9 13 ,5

14

1 i , 4

2 2 ,8

3 4^2

-1 5, 6

5 : P6 0,4

7 v .*

8 11 ,2

9 12 6

2B5.

2 8 6

2 87

28 8

2 8 9

484 500 515 530 54S

637 652 6O7 682 697

788 803 8T

8 834 849

9 39 9 5 4 9 6 9 9 8 4 *

46 09 0 105 120 13 5 15 0

561 57 6 591 606 621

712 728 743 758 773

864 879 894 90 9 9 2 4

rOi $ ( ,030 *C45 o(-o *o7 5

165 180 195 210 225

18

1 J,j82

3 ,6

3 5 /4

4 7 , 2

5 9 , o

6 n i,8

7 1 2 , 68 14 ,4

9 16 ,2

17

1 1,7

2 3 , 4

3 5 ,1

4 6 ,85 8 ,5

6 1 0 , 2

7 1 1 9

8 15 ,6

9 1 5 3

161 1 ,6

2 . 2

3 4 ,8

4 6 , 45 3 ,o

6 9 ,6

7 XI 2

1^,3

9 1 4 , 4

15

1 1 5

2 3 ,

3 4 5

4

5 7 5

9 , o

7 i e / S

8 12 ,0

9 13 ,5

14

1 i , 4

2 2 ,8

3 4^2

-1 5, 6

5 : P6 0,4

7 v .*

8 11 ,2

9 12 6

29029 1

2 9 2

2C.3

2 9 4

240 255 ^70 28^ 300

3 8 9 4 0 4 4 1 9 4 3 4 4 4 9

53* S i3 568 583 59 8

637 702 710 731 746

835 8^0 864 879 894

315 33 35 9 37*'

4 6 4 4 7 9 4 9 4 5 9 5 2 3

013 627 642 057 672

7 f t i 7 7 6 j'.p 8o,5 82 0

909 w23 "3s

9 5 3 9 6 7

18

1 J,j82

3 ,6

3 5 /4

4 7 , 2

5 9 , o

6 n i,8

7 1 2 , 68 14 ,4

9 16 ,2

17

1 1,7

2 3 , 4

3 5 ,1

4 6 ,85 8 ,5

6 1 0 , 2

7 1 1 9

8 15 ,6

9 1 5 3

161 1 ,6

2 . 2

3 4 ,8

4 6 , 45 3 ,o

6 9 ,6

7 XI 2

1^,3

9 1 4 , 4

15

1 1 5

2 3 ,

3 4 5

4

5 7 5

9 , o

7 i e / S

8 12 ,0

9 13 ,5

14

1 i , 4

2 2 ,8

3 4^2

-1 5, 6

5 : P6 0,4

7 v .*

8 11 ,2

9 12 6

29 5

2 9 6

29 7

2 98

2 9 9

982 997 $0 12 *02fi *C4I

47 129 1 44 159 7 3 18 8

27^ 2 90 305 3111 33 4

4 22 43 6 451 46 5 4*'

5^ 7 .182 5 9 6 6 1 1 6 2 5

*05 6 1 .070 *U85 , ,100 #114

202 2 ' 7 232 246 261

349 36

3 378 3 9* 4


21/39


L 0 1 2 3 4 5 6 7 3 9 f P.

300 4 7 - 1 2 7 * 7 7 4 1 7 5 6 7 7 7 8 4 7 9 9 8 1 3 8 2 8 8 4 23

C I 8 5 7 8 -1 8 8 5 9 0 0 9 1 4 9 29 9 4 3 9 5 8 y 7 2 9 8 6

3 " 48 OCI ,15 G2 , c 44 0 5 8 7 3 1187 l i b " 3

3C

3 14 4 159 173 187 20 2 2 1 6 2 1 0 2 4 4 25Q 2 7 3 1 fi2 8 7 3 0 2 31 b 3 3 ^ 344 359 37 i 38 7 4 0 I 4 1 6

1

1 0 )

I c

3 5 4 3 4 4 4 58 4 7 3 4 8 7 5 i 5 i 5 5 3 o 5 4 4 5 582 3

3 0 0 5 7 2 5 8 6 Go i 6 1 5 6 2 9 6 4 3 65 7 b 7 I 6 8 6 7 0 0 3 4 5

3 7 7 1 4 " 2 8 7 t 2 , 7 5 7 7 0 7 ^ 5 7 9 9 8 1 3 02 7 841 46,0

3O 8 5 5 8(39 S 8 3 8y 7 9 " 9 2 b 9 4 0 9 5 4 96 B 9 8 2 5 7 5

3 ? 9 9 6 , 0 11 1 * 0 2 4 . 0 3 8 , 0 5 2 * o 6 6 . U B O O y 4 , 1 0 8 . 1 2 267 9 ,

1 0 , 5

31 0 4 9 1 3 6 IS O 164 178 192 20 6 22 1 2 3 4 2 4 8 2 6 28 12 0

3 " 2 7 0 2 9 0 3< 4 3 1 8 3 3 2 3 4 b 3 3 7 4 4 0 29

J3 , 5

3 1 a 4 1 5 4.19 4 4 J 4 5 7 4 7 1 4 8 5 49*- 5 1 3 5 2 7 5 4 1

3 * 3 5 5 4 S b 8 5 8 2 5 9 6 Q i o 6 2 4 6 38 6 5 1 6 6 5 6 7 9

314 b93 7 0 7 7 2 1 7 3 4 7 4 8 7 6 2 7 7 0 7 9 0 8 o j 8 1 7

3 1 5 8 3 1 8 4 5 8 5 9 8 7 2 8 8 6 9 0 0 9 1 4 9 2 7 9 4 1 9 5 ? 143 1 6 9 6 9 9 8 2 9 9 6 * o i o . 0 2 4 0 3 7 * 5 I * b ; , 0 7 9 X) 2 1 1 4

31 7 50 106 120 133 147 1 61 17 4 18 8 2 0 2 2 i i 22 y 2 2 ,83 1 8 2 4 3 2 5 b 2 7 0 2 8 4 29 7 3 T - 3 2 ? 3 3 8 3 5 2 3 6 5 3 4 2

3 1 9 3 7 9 3 9 3 4 0 6 4 2 0 4 3 3 4 4 7 4 b i 4 7 4 4 8 8 5 I \ 5 632 0 51 5 5 2 9 5 4 2 5 5 6 5 6 9 5 8 3 5 9 6 6 1 0 6 2 3 63 7 |

7,o8 . 4

3 2 1 6 5 1 6 6 4 6 7 3 6 9 1 7 5 7 1 8 7 3 2 7 4 5 7 5 9 7 7 2 7 9 , 83 2 2 7 8 6 7 9 9 8X 3 8 2b 8 5 3 8 6 0 8 8 0 8 9 3 8 1 1 , 2

32 3 9 3 4 9 4 7 Q 6 I 9 7 4 * o o i . 0 1 4 , 0 2 8 4 I 9 1 2 . 6

32

4 5 I 0 5 5 0 68 0 8 1 9 5 i Of ' 12 1 1 3 5 1 4 8 16 2 1 75

1 2 . 6

3 2 s 1 8 8 2 0 2 2 i 5 2 2 8 2 4 2 25 5 2 68 2 8 2 2 95 3 0 8

3 2 b 3 2 2 3 3 5 3 48 3 6 2 37 5 38B 402 428

32 7 45 b 468 +81 4 9 5 50 8 5 2 1 5 3 4 54 8 5 6 I 5 7 4 1 132 8 58 7 6 0 1 6 1 4 6 2 7 6 4 0 6 5 4 6 6 7 b * o f 9 3 7 0 6

32

y 7 2 0 7 3 3 7 4 b 7 5 9 7 7 2 7 8 6 7 9 9 8 1 2 8 2 ; 312

1,32 6

33 0 851 8bJ 8 7 S 89 1 9 0 4 9 1 7 9 3 9 4 3 9 5 7 9 7 0 3 3 , 9

3 3 i 9 8 3 9 9 b , 0 0 9 * 0 2 2 . 0 3 5 * 4 8 * o 6 i , 0 7 * , 0 8 8 . 1 0 1 4 5 , 2f. r-3 3 2 5 2 " 4 12 7 1 40 15 3 1 5 5

T7 9 1 92 2 0 5 2 1 8 2 3 1 5 5

3 3 3 2 4 4 25 7 2 7 0 2 8 4 2 9 7 3 1 0 3 2 3 3 3 6 3 4 9 3 6 26 7 J

3 3 4 3 7 5 388 401 414 4 2 7 44 0 4 5 3 4 6 6 4 79 4 9 2 ,7 9 , 11 0 , 4

3 3 5 5 0 4 5 1 7 5 3 5 4 3 5 5 6 5 6 9 5 8 . 5 9 5 6 0 8 6 2 19 " 7

33


22/39

LOGARITHMIC TABLES27

N L 0 1 2 3 4 5 6 7 8 9 P P

35 0 5 4 4 0 7 4 1 9 4 3 2 4 4 4 4 5 6 4 6 9 4 8 1 4 9 4 5 0 6 5183 5 1 531 543 55 .? 5 6 8 5 8 0 5 9 3 box 6 1 7 6 3 0 0 4 23 5 2 6 5 4 6 6 7 ^ 7 9 6 9 1 7 0 4 7 1 6 7 2 8 7 4 1 7 5 3 7 6 S

3 5 3 7 7 7 7 9 0 8 0 2 8 x 4 8 2 7 8 3 9 85 X 804 87 6 88 8

3 5 4 9 0 0 9 1 3 9 2 5 9 3 7 9 4 9 9 6 2 9 7 4 9 8 6 9 9 8 S .OI I13

3 5 5 5 5 0 2 3 03i 0 4 7 0 6 0 0 7 2 0 8 4 C 9 0 1 0 8 I 2 1 T 3 32

1 32 0

35f> 1 4 $ 15 7 1 6 9 1 8 2 1 9 4 2 0 6 2 1 b 2 3 c 2 4 2 2.5S 3 i , s

3 5 7 2 6 7 2 7 9 2 9 1 3 " 3 3 ^ 3 2 8 3 4 c 3 5 2 3 6 4 3 7 6 4 5 , 2

3 5 f 3 8 8 4 0 0 4 1 3 4 2 ? 4 3 7 4 49 4 6 1 4 7 3 4 8 5 4 9 7 5 6 , 5

3 5 9 5 0 9 5 2 2 5 3 4 5 4 6 "5 8 5 7 0 5 8 2 5 9 4 6 0 6 6 1 867 7 ?

9 / 1

36 0 6 3 0 6 4 2 6 5 4 6 b 6 6 7 8 6 0 1 7 C 3 7 1 ? 7 2 7 7 3 05 1QA

3 6 1 7 5 ' 7 6 3 7 7 5 7 8 7 7 9 9 b i i 8 2 3 8 3 5 8 4 7 8 5 9 9 1 1 / 7

3 6 2 87 1 8 8 3 8 9 5 9 1 8 2 1 9 4 2 0 5 2 x 7

3 6 5 2 2 9 2 4 1 2 5 3 26 5 2 7 7 2 8 9 3 0 1 3 1 2 3 2 4 3 3 6 123 6 6 3 4 8 3 0 0 3 7 2 3 8 4 3 9 6 4 7 4 1 0 4 3 1 4 4 ? 4 5 5 I 1 23 6 7 4 6 7 4 7 8 4 9 0 5 0 2 5 * 4 5 2 6 5 3 8 5 4 9 5 6 1 5 7 3 2 2 A3 6 8 5 8 5 5 9 7 6 0 8 O 2 0 6 3 2 6 4 4 6 5 6 6 6 7 6 7 9 0 9 1 3 63 6 9 7 3 7 1 4 7 2 0 3 * 7 5 0 7 6 1 7 7 3 7 8 5 7 9 7 bo 8

3

t 4 8

37 0 8 2 0 8 3 2 8 4 4 8 5 5 8 6 7 8 7 9 8 9 1 9 0 2 9 1 4 9 2 656

6 0

7 , 2

3 7 i 9 3 7 9 4 9 9 6 1 9 7 2 9 8 4 9 0 6 ji,CO8 *oi9 *3I 43 7 i 4

3 7 2 5 7 0 5 4 0 0 6 0 7 8 0 8 9 10 1 113 1 2 4 1 3 6 1 5 9 8 9 , 63 7 3 1 71 1 8 3 1 94 2 0 0 2 1 7 2 2 9 2 4 1 252 2O4 276 9 10 a

3 7 4 2 8 7 2 9 9 3 1 0 3 2 2 3 3 4 3 4 $ 3 5 7 3 0 8 3 8 0 3 9 2

3 7 5 4 3 4 1 5 4 2 6 4 3 8 4 4 0 4 6 : 4 7 3 4 8 4 4 9 6 5 7

3 7 6 519 5 3 5 4 2 5 5 3 5 6 ? 5 7 6 S8 8 o o c 6 1 1 0 2 33 7 7 0 3 4 6 4 6 6 5 7 6 0 9 6 S 0 6 9 2 7 0 3 715 7 2 6 7 3 8 113 7 8 7 4 9 7 0 1 7 7 2 7 8 4 7 9 5 8 0 7 8 1 8 8 3 0 fs4i B 5 23 7 9 b- H 8Y S 8 8 7 8 9 8 9 1 0 9 2 1 9 3 3 9 4 4 9 5 5 9 0 7 1

2

x 1

2

38 0 9 7 8 9 9 0 ooi 0 1 3 , 0 2 4 , 0 3 4 , , 0 4 7 # 0 5 8 # 0 7 0 * o 8 i 3 ?. 338 : 5 8 C 9 2 10 4 1 2 7 1 3 8 1 4 9 16 1 . 7 2 1 8 4 19? 4 4 , 43 8 2 2 C6 2 1 S 2 2 9 2 4 c 2 5 2 2 6 3 2 7 4 2 8 6 2 9 7 3 9 5 5 , 5

3 8 3 3 2 0 3 3 1 .343 3 5 4 3 6 S 3 7 7 6 1 1 8 1 2 9 I JO 151 1 6 2 1 7 3 1 8 4 I9S 2 0 7 1

1U

1 , 03 9 i 2 1 8 2 2 9 2 4 c 2 5 1 2 6 2 2 7 3 2 8 4 2 9 5 3 0 6 318 2 2jO3 9 2 3 2 9 3 4 r 3 5 1 3 6 2 373 3 8 4 3 9 $ 4 C6 41 7 , 2 8 3 3 / 393 4 3 9 4 5 4 0 1 4 7 2 483 4 9 4 5 0 6 517 5 3 9 d. 4 / 3 9 4 5 S 0 5 6 I 5 7 2 5 8 3 5 9 4 t o j 6 1 6 6 2 7 6 3 8 0 4 9

5 5 /

3 9 5 6 6 0 6 7 1 6 8 2 0 9 3 7 0 4 715 7 2 6 737 7 4 8 7 5 96

7

6 , 0

7P39t> 7 7 0 7 8 0 7 9 1 8 0 2 8 1 5 8 2 4 8 3 5 8 4 6 8 5 7 8 6 8 8 8,o

1

3 9 7 8 7 9l 9 C 9 0 1 9 1 2 9 2 3 934 9 4 ? 9 5 0 9 0 6 9 7 7 9 9,o

398 9 8 86 0 0 9 7

9 9 9 * O I O * 02 X *


23/39


N I. 0 1 2 3 4 5 6 7 8 9 P P.

40 0 60 206 21 7 228 2 3 9 2 4 9 2 b 0 27 1 2 8 2 2 9 3 3 44 0 1 3 1 4 325 33> 347 3 5 8 3 6 9 3 7 9 3=53 yi 3 o 7 3 0 8 3 0 9 4 104 114 124 1 3 442 B 14 4 15 5 16 3 1 75 1 85 ! 9 5 205 215 2 2 5 2 3 b

78

759 6 7 4 b 8 9 6 9 9 7 ; 7 1 9 7 2 4 7 3 4

4 3 4 7 4 9 7 5 9 7 6 9 7 7 9 7 8 9 7 9 9 S09 819 b2f 8 3 9

4 3 5 8 4 9 8 3 9 8b ) 8 79 889 8 9 9 9 y 9 1 9 9 2 9 9 3 9

4 3 b 9 4 9 9 5 9 9 7 9 9 8 8 9 9 8 * o i x G2 8 * 3 8 94 3 7 6 4 0 4 8 0 5 8 0 6 8 0 7 8 0 = 8 C1 8 108 118 128 13 7

438 147 157 IB7 17 7 18 7 19 7 207 217 227 2 ? 7 I f 9

4 3 9 2 4 6 2 5 6 2 bb 27b 2bb 2(p 30ft 31' 3 2 b 3 3 52

3

1 8

2 . 7

44 0 3 4 5 3 5 5 36 5 3 7 5 3 8 5 3 9 5 4 0 4 4 1 4 4 2 4 4 3 4 i3 . 6^ C

44 1 4 4 4 4 54 44 4 7 3 4 8 3 4 9 3 5 3 5 1 3 5 2 3 5 3 25g 4 , 5

4 4 2 5 4 2 55 2 562 5 7 2 5 8 2 5 9 1 6crr 6 1 1 6 2 1 6 3 15 / 4f. ~

4 4 3 b 4 0 6 5 0 6 6 0 6 7 0 b 8 o 6 8 9 6 9 9 7 0 9 - 1 9 / 2 b7g

4 4 4 7 3 8 7 4 8 7 5 8 7 0 8 7 7 7 7 7 7 9 7 8c 7 - l b B2b9

7 ,2

8 , 1

4 4 5 8 3 6 8 4 6 8 5 6 8 6 5 87 5 8 8 ? 8 9 3 9 0 4 9 1 4 9 2 44 4 6 9 3 3 9 4 3 9 5 3 9 6 3 9 7 2 0 8 2 992 OO2 ,011 *02I4 4 7 6 5 3

r 0 4 0 0 5 0 c f o 0 7 0 07Q 089 c 9 9 10 8 1184 4 8 [ 2 8 "37 '4 7 157 167' 176 186 196 205 2 1 3

4 4 9 2 2 3 2 3 4 244 2 5 4 2 6 3 2 7 3 2 8 3 2 9 2 3 0 2 3 1 2

4303 2 1

3 3 1 3 4 i 3 5 3 6 0 3 6 9 3 7 9 3 8 9 3 4 84 0 8

N. L 0 1 2 3 4 5 6 7 8 9 p ?


24/39

LOGARITHMIC TABLES29

N L. 0 1 8 3 4 5 6 7 8 9 P P

45 04 ? i

4 5 *

4 5 3

4 5 4

6 5 3 3 i 3 4 i 3 5 3 'JO

418 (27 437 {47 456

5 1 4 5 2 3 5 3 3 5 4 3 5 5 2

b i t . b i q 6 2 9 6 3 9 6 4 8

7c S 71 - 72 5 7 14 7 44

369 379 389 398 4 'J8

466 475 485 495 504

5 6 2 5 7 x 5 8 1 5 9 1 6 0 0

6 5 8 6 6 7 6 7 7 6 8 * 6 9 6

7 5 3 7 6 3 7 7 * 7 8 2 7 9 2

0

1 1, 02 J O

3 3,4 4 ,b

5 5 , 6 6 ,0

7 7 , 8 8 ,0

9 9."

9

1 0 , 92 1,83 * ,7

4 3 , 6

5 4 ,5

6 5 , 4

7 6 , 3

8 7 29 1

8

1 0, 82 1,6

3 * ,4

4 3 ,2

5 4 , 6 4 s

7 5 ,68 6 ,49 7 3

4 5 5

4 5 6

45 74 5 8

4 5 9

ao i 8 11 820 83 0 839

896 906 9*6 92 .5 935

9 9 2 j c o l , 0 1 1 * 7 2 7 6

3*9 338 3 4b 35$ 3b

41

7 4*5 434 443 45*

0

1 1, 02 J O

3 3,4 4 ,b

5 5 , 6 6 ,0

7 7 , 8 8 ,0

9 9."

9

1 0 , 92 1,83 * ,7

4 3 , 6

5 4 ,5

6 5 , 4

7 6 , 3

8 7 29 1

8

1 0, 82 1,6

3 * ,4

4 3 ,2

5 4 , 6 4 s

7 5 ,68 6 ,49 7 3

4 9 5

4 9 6

4 9 7

4 9 8

4 9 9

46 '- 469 47 8 487 490

548 55; 56b 57 4 -3

636 b44 653 662 677

7 * 3 7 3 * 7 4 9 7 5 8

SID


25/39

26 No. 5 3 U S E OF LOGARITHMS

N L 0 1 2 3 4 5 6 7 8 9 I P

500 6 9 9 7 9 0 0 9 1 4 9 2 3 9 3 2 9 4 c 9 4 9 9 5 8 9 6 b 9 7 55 0 1 0 8 4 9 9 2 * o o i o i o o i 8 - , 0 2 7 . 0 3 6 * 04 4 , 0 5 3 C b2

5 0 2 7 0 0 7 0 0 7 4 o o 8 0 9 b 1 0 3 " 4 1 2 2 1 3 1 1 4 0 4 8

5C

3 15 7, i6i 1 7 4 - 8 3 I y l 2 0 0 2 0 9 2 1 7 2 2 b 2 3 45 4 24 : 2 5 2 2 bO 2 6 y 2 7 8 2 8 6 2 9 5 3 3 3 1 2 3 2 1

5 5 3 2 9 3 3 8 3 4 6 3 5 5 3 6 4 3 7 2 3 8 i 3 8 9 3 9 8 4 0 6

50 & 4 i 5 4 2 4 4 J 24 4 1

4 4 9 4 5 8 4 b 7 4 7 5 4 8 4 4 4 2 *5 Z 5 0 1 5 1 8 5 2 6 5 3 5 5 4 4 5 5 2 S b i 5 6 9 5 7 8

*

S 5 8 6 5 9 5 0 0 3 6 1 2 6 2 1 6 2 q 6 3 8 6 4 b 6 5 5 6 6 3 *' 9

5 9 6 7 2 0 8 0 6 8 9 6 9 7 7 0 b 7 r 4 7 2 3 7 3 1 7 4 0 7 4 923

1 , 8

2 , 7

51 0 7 5 7 7 6 6 7 7 4 7 8 3 7 4 1 8 0 c 8 0 8 8 1 7 S 2 5 8 3 41

5 " S 4 2 8 5 1 3 5 . , 3 6 8 8 7 6 8 8 5 8 9 3 9 3 2 9 1 0 9 1 956

5 1 2 9 2 7 9 3 5 9 4 4 9 5 2 9 0 1 9 6 * 9 7 8 9 8 6 9 9 5< 45 4- s

5 1 3 7 i 0 1 2 0 2 0 0 2 9 0 3 7 4 6 5 4 0 5 3 37 1 0 7 0 0 8 8/

8b . 3

5 1 4 O y 6 1 0 3 " 3 1 2 2 1 3 0 1 3 9 1 4 7 1 5 5 1 6 4 1 7 29 ft

5 1 5 i i i 1 8 9 1 9 8 2 C b 2 1 4 2 2 3 2 3 1 2 4 0 2 4 8 2 5 75 1 6 2 6 5 2 7 3 2 S 2 2 9 0 2 0 9 37 3 1 5 3 2 4 3 3 2 3 4 1

. " 7 3 4 9 3 5 7 3 6 6 3 7 4 3 8 3 3 9 * 3 9 9 4 0 ? 4 1 6 4 2 55 1 8 4 3 3 4 4 i 4 5 0

4 5 8 4 6 6 4 7 5 4 8 3 4 9 2 < 08

5 1 9 5 1 / 5 2 5 5 3 3 5 4 2 5 5 5 5 9 5 6 7 5 7 5 5 8 4 5 9 2

520 b o o 0 0 9 6 1 7 6 2 5 6 3 4 6 4 2 6 5 0 6 5 9 6 6 7 6 7 55 2 1 6 8 4 6 9 2 7 0 0 79 7 1 7 7 2 3 7 3 4 7 4 2 7 } o 7 5 9 $5 2 2 7e7 7 7 5 7 8 4 7 9 2 So o 8c ^ 8 1 7 8 2 5 3 4 S 4 25 2 3 b i o 8 5 8 8 6 7 8 7 5 8 8 3 8 9 2 9 0 0 9 0 S 9 1 7 9 2 3

1 0 , 8

5 2 4 9 3 3 9 4 1 9 5 9 5 8 9 0 6 9 7 5 9 8 3 4 9 1 9 9 9 , 0 0 02

3 2 . 4

5 2 5 7 2 0 1 6 0 2 4 0 3 2 0 4 1 0 4 9 0 5 7 o b 6 0 7 4 0 8 2 0 9 04 3 : 2

5 2 6 0 9 9 1 ; " 5 1 2 3 1 3 2 1 4 0 1 4 8 1 5 b 1 6 5 1 7 35 4 ,

&

1 85 2 7 1 8 1 1S9 1 & 2 0 b 2 1 4 2 2 2 2 3 " 2 3 9 2 4 7 2 3 5 4 ,- f5 2 b 2 t >3 2 7 2 2 8 c 2 8 8 2 9 b 3 U 4 3 U 3 2 1 3 2 9 3 3 7

78 .

S 2 9 3 4 0 3 5 4 3 0 2 3 7 b 3 7 3 8 7 3 9 5 40 5 4 1 1 4 1 99 7 , 2

53 0 4 2 8 4 3 6 4 4 4 4 5 2 4 b o 4 6 9 4 7 7 4 8 ? 4 9 3 5 0 1S 3 i 5 9 y - * 5 2 6 5 3 4 5 4 2 5 5 b 5 5 8 5 6 7 5 7 5 5 8 3

5 3 2 5 y i 5 9 9 6 0 7 6 1 6 0 2 4 6 3 2 0 4 0 0 4 8 6,5 6 6 6 3

&5 5 3 6 7 36 8 1 6 8 9 6 9 7 7 0 5 7 1 3 7 2 2 7 3 b 7 3 8 7 4 6 w

5 3 4 7 5 4 7 6 2 7 7 0 7 7 9 . 7 8 7 7 9 5 . 8 0 3 i l l* 8 1 9 8 2 7

5 3 5 8 3 5 3 4 3 8 5 2 8 b o 8 b 8 8 7 6 8 8 4 8 9 2 o o o 9 0 8

53> 9 1 6 9 2 5 9 3 3 9 4 1 9 4 9 9 5 7 0 6 5 9 7 3 9 8 1 9 8 9 75 3 7 9 9 7 , 0 1 4 U 2 2 , 0 3 0 ,0 38 . ,040 54 +7

5 3 8 7 3 0 7 8 o b b 0 9 4 1 0 2 i l l " 9 1 2 7 1 3 5 14 3 1 5 11 4 , 7

5 3 9 1 5 9 1 6 7 1 7 5 1 8 3 1 9 1 1 9 9 2 0 7 2 1 3 2 2 3 2 3 12

3 2 1

540 2 3 9 2 4 7 2 5 5 2 6 3 2 7 2 2 8 0 2 8 8 2 0 6 3 0 4 3 1 22 8

5 4 i 3 2 0 3 2 8 3 3 f 3 4 4 3 5 * 3 6 0 3 6 b 3 7 6 3 P 4 3 9 2 X3 5

? 4 2 4 0 0 4 0 8 4 1 b 4 2 4 4 3 2 4 4 4 4 8 4 5 6 4 4 4 7 20 4 2

5 4 3 4 * 0 4 8 8 4 9 b 5 4 5 2 0 5 2 8 5 3 " 5 4 4 5 5 27Q

4 . ^r h.

5 4 4 5 6 0 5 6 8 5 7 6 5 8 4 5 9 2 b o o 6 0 8 6 1 6 b 2 4 6 3 2O

9

5 b

6 , 3

5 4 c 6 4 0 6 4 8 6 5 6 6 6 4 b 7 2 6 7 9 6 8 7 6 9 5 7 3 7 1 1

5 b

6 , 3

5 4 " 7 1 9 7 2 7 7 3 5 7 4 3 7 5 1 7 5 9 7 6 7 7 7 5 7 8 3 7 9 i

5 4 7 7 9 9 8 0 7 8 1 5 8 2 3 8 3 0 8 3 S 8 4 " 8 5 4 X 0 2 8 7 0

3 + 8 8 7 8 8 S b 8 9 4 9 0 2 9 1 0 9 1 8 9 2 6 9 3 3 9 4 1 9 4 o

5 4 9 9 5 7 9 ^ 5 9 7 3 9 8 1 9 8 9 9 9 7 * 5 * O I 3 *'J2Q < 0 2 8

55 0 7 1 0 3 6 0 4 4 0 5 2 c 6 o 0 6 8 0 7 6 0 S 4 C 9 2 9 9 1 07

H, L 0 1 2 3 4 5 6 7 8 9 P P


26/39


M i . 0 1 2 3 4 5 6 7 8 9 P P

55 05 3 1

5 5 *

5 5 3

5 5 4

74 C36 04 4 0 52 c5 o c6 8

1 1 5 1 2 3 i j i 13 9 14 7

I 9 4 202 2IO 21? 22 5

273 280 288 29(3 3 O.,

35* 35 9 37 374 3 S 2

076 084 092 009 107

153 Ib2 170 178 I 6

2 3 3 241 249 257 2 6 3

312 320 327 335 343

390 398 406 4 1 , 42 1

8

1 0 ,?

2 1 6

3 * , 44 3-2

5 4 - e

6 4,8

7 5 -68 6,4

9 7 - 2

7

1 0, 72 1 4

3 2. 1

4 2 8

5 3 -56 4 2

7 4 , 9

8 5 '

9 6 , 3

5 5 5

5 5 6

55 7

5 58

5 5 9

429 437 445 4 5 3 4 M

5 7 5r

0 5 2 3 S 3 ' 5 3 9

586 593 h o i 009 017

663 67 1 079 687 693

741 749 757 7&4 772

468 476 484 492 500

5 4 7 5 5 4 5 7 5 78

b 24 032 640 648 656

702 710 71 c 72b 7 3

780 788. 79 6 803 8 1 1

8

1 0 ,?

2 1 6

3 * , 44 3-2

5 4 - e

6 4,8

7 5 -68 6,4

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