PRIRCE 
 
GIFT or 
 
 lirs. G. N, LeTTis 
 
 t 
 
 
SHORT TABLE OF INTEGRALS 
 
 BT 
 
 B. O. PEIRCE 
 
 HoLLis Professor of Mathematics and Natural Philosophy in 
 Harvard University 
 
 SECOND REVISED EDITION 
 
 GINN AND COMPANY 
 
 BOSTON • NEW YORK • CHICAGO • LONDON 
 ATLANTA • DALLAS • COLUMBUS • SAN FRANCISCO 
 
COPYRICIIT, ISVd, 1910 
 
 By GINN and COMPANY 
 
 ALL EIGHTS RESERVED 
 522.2 
 
 1 
 
 Since I cannot hope that these formulas are wholly free 
 from misprints, I shall be grateful to any person who will 
 call my attention to such errors as he may discover. 
 
 B. O. PEIRCE, 
 Harvard University, Cambridge. 
 
 GIFT 
 
 tH)t Stfjenatum jPrees 
 
 I'.IW" AMI CllMl'ANY • Vlli). 
 PKIETURS • liUSTuN • U.S.A. 
 
Q A- =s / o 
 HA If ' 
 
 TABLE OF INTEGRALS. 
 
 -o-o>0<o°- 
 
 I. FUNDAMENTAL FORMS. 
 
 1. I adx = ax. 
 
 2. \ ctf(x) dx = a ( f(x) dx. 
 
 3. f— = logx. noga: = log(-x) + (2A:+l)7r/.] 
 J x 
 
 oc'^dx = ' when vi is different from — h 
 
 m -\- 1 
 
 5. Ce'^dx = e'. 
 
 6. C a^ log adx = a", 
 
 7 I _^^- — = tSLU-^x, or — ctn-^a;. 
 
 . = sin~^cc, or — cos ^x 
 
 dx 
 
 / dx 
 X Va;^ — 1 
 
 10. . , 
 
 V2 X — X 
 
 = sec-^cc, or — esc ^x. 
 -1 
 
 dx 
 
 — = versin-^a;, or — coversin-^a;. 
 
 -J'2x-x^ 
 
 M623267 
 
FUNDAMENTAL FORMS. 
 
 1. I COS xdx = sin x, or — coversin x. 
 
 2. I ^va.xdx = — cos x, or versin x. 
 
 3. I ctn ic c?x = log sin x. 
 
 4. I tan xdx = — log cos x. 
 
 5. I tan X sec a; (fx = sec 05. 
 
 6. I sec^icc?ic = tancc. 
 
 7. I csc^ajt^x = — ctnx. 
 
 In the following formulas, u, v, w, and y represent any 
 unctions of x : 
 
 8. i (u-\-v + w + etc.) dx = i udx + i vdx + i wdx-\- etc. 
 
 9 a. i udv ^= uv — i V du. 
 
 _ /• c?y , f du ^ - ' 
 
 9 6. \ u—-dx = uv — \ v-r- dx. 
 J dx J dx 
 
 .o.//(.)^=//Mf^ 
 
 dx 
 
RATIONAL ALGEBRAIC FUNCTIONS. 
 
 II. RATIONAL ALGEBRAIC FUNCTIONS. 
 
 A. — Expressions Involving (a + hx). 
 
 The substitution oi y ov z for x, where y = a -\- bx, 
 z = (a + bx) / X, gives 
 
 21. ("(a + bx)^dx = ijy^dy. 
 
 22 
 23 
 
 . Cx{a + bxYdx = y,yf {y - «) dy- 
 
 r x'^dx ^ _i_ r {y-(^Tdy . 
 
 ^ J Ca + Z^x)"* i»+V 
 
 25 
 
 (a + Z^x)"* i"+V 2/'" 
 
 ■J a;«(a4-te)'" «""+"- V 
 
 Whence 
 
 dx 1 
 
 (Zx _ 
 
 + bxf " 
 
 dx 1 
 
 27. fy- 
 
 J (a 
 
 r dx ^ 1 
 
 ^^- J (a + ix)« 2 6(a + te)= 
 
 
6 RATIONAL ALGEBRAIC FUNCTIONS. 
 
 r ^^ix _ 1 r 1 a ~| 
 
 J {<i + te)' ^z-' L « + ^* 2 (a + bxy\ 
 
 32. /;f^ = ^3 \i (* + ^^)' - 2a (a + hx) + a" log (a + bx)\ 
 
 34, 
 
 35 
 
 / dx _ 1 ■■ a + ^>x _ * 
 a; (a + 6j:;) a x 
 
 /dx _ 1 _ \_ , rz-t-^A-r 
 xfa+^a;)^ a(a+bx) a^ 
 
 (a+bxf a{a+bx) a^ x 
 
 37. C (a+bxY {a'+b'x)'"dx = - — ■ — -y ( (a+bxy + ' (a'+b'xf 
 
 J ^ ^ ^ ' {in-\-n-\-\) b \ 
 
 — m(ab'-a'b) C (a-^-bx)" (a' + b'xy-hlxj 
 
 r {a + bxydx 1 f (a + hTY + ^ 
 
 ' J (a' 
 
 I _L /,' ,• \in — L 
 
 {a' + b'x)'" (m — l){ab' - a'b) \{a' + b'x) 
 
 r(a + bxYdx 
 + (m — n — Z)b I -r-; z-^: r 
 
 ^ 1 / (a-\-bxY ' 
 
 ~ (m — n-l)b'\(a' + b'xy"-^ 
 
 + n 
 
 ^ ^J {a' + b'x)'" J 
 
 _ i f (a + bx)" r (a + bx)''^^dx \ 
 
 * /'—^^^— =--'- + - log ^^i-^ . 
 J x\a + bx) ax a"^ " x 
 
39 
 
 RATIONAL ALGEBRAIC FUNCTIONS 
 
 dx 1 , ft' + Vx 
 
 (a + bx) {a' + h'x) ~~ ab' -a'b' °^ a-\- bx 
 
 dx 
 
 40./ 
 
 (a + bxy {a' + b'x)"" 
 
 = I ( 1 
 
 (m - 1) (ab' - a'b) \{a + bx)"-^ (a' + b'x)"'-' 
 
 dx ^ 
 
 41 
 
 - (m + n-2) bC 
 
 f 
 
 (a + bxy (a' + b'xy"-\ 
 
 xdx 
 (a + bx) (a' + b'x) 
 
 42, j 
 
 43 
 
 (a + bx)\a' + b'x) 
 
 = _J— f^- + — ^— log ^^^±^\ 
 
 ab' — a'b \a + bx ab' — a'b a + bx J 
 
 r xdx 
 
 (CI + bxf (a' + b'x) 
 
 — a a' . a' + b'x 
 
 TTTz log ■ 
 
 b (ab' - a'b) (a + bx) (ab' - a'by ^ a + bx 
 
 x'^dx a^ 
 
 (a + bxy (a' + b'x) ~~ b' (ab' - a'b) (a + bx) 
 
 1 r«'% / r . 7-r N , a(ab'-2a'b) ^ . , . .1 
 — ^ [y log (a' + b'x) + -^—j, '- log (a + bx)j 
 
 /I n 2+i 
 
 "^ («&' - 
 
 .^ C dx n , ,-. ^~ 
 
 (a + OXJn ^ ^ 
 
8 RATIONAL ALGEBRAIC FUNCTIONS. 
 
 B. — Expressions Involving (a + 5x"). 
 
 r dx 1, _iX 1 . _, X 
 
 47. I -T", — -„ = -tan ^-=-sin i— ===• 
 
 .^ C dx It c + x C dx 1 . .-r — c * 
 
 48. I -^ 5 = — log > \ —^ i = 77-log — — -• 
 
 J c^ — x^ 2 c c — xjx^ — cr 2 c x-\-g 
 
 50. I — r^-i = — 7=log--= 7=j if a>0, b<0. 
 
 J a + bx^ 2V^^ V^-a;V-6 
 
 /* <?a; _ a; 1 /* c?x 
 
 J (a + bxy ~ 2a(a + bx') 2aJ a + bx^' 
 
 r dx _ 1 ^ _i_ ^ ^^ ~ 1 r rfa 
 
 ■ J (a + Z/x2)"*+i - 2 7/ia (a + bx^)"" 2 ma J {a + bx^)"^ 
 
 " ./ a; ((X + 6x^) 2 a a + bx'^ 
 
 /x^dx _x a r dx 
 a + bx^ b bJ a + bx^ 
 
 56 
 
 r dx _ _ j_ _ ^ r_^_ 
 
 'J cc^ (a + ix'^) ax a J a + 6x^ 
 
 , a;^c?aj — x . 1 /* r/.r 
 
 58. 
 
 r x^dx _ — X '^ C 
 
 (a -I- bxy'' + ^ 2mb(a + bx^"" 2 mbJ (a + bx^)'» 
 
 r dx -^ c ^^ ^ r— -^^— ■ 
 
 J x\a + ^<x2)'» + i ~ a J x'^{a + ix^)*" a J (a + 6x')'"+*" 
 
 • p dx 1 
 
 y^-s-"--©^ /i^.-i-."-©- 
 
RATIONAL ALGEBRAIC FUNCTIONS. 
 
 r« 
 
 -ii 
 
 >« 
 
 -«) 
 
 (U 
 
 <o 
 
 Si 
 
 u 
 
 a> 
 
 <o 
 
 rSli 
 
 rS, 
 
 ^ 
 
 ^ 
 
 '- 
 
 •^ 
 
 1 
 ( 
 
 1 
 
 1 
 
 1 
 
 1 
 
 r-* 
 
 1 
 
 •'« 
 
 a 
 
 c« 
 
 05 
 
 -t-3 
 
 -1-3 
 
 ^ 
 
 ^ 
 
 H 
 
 h 
 
 H 
 
 h 
 
 
 
 <ii e 
 
 H 
 
 8 
 + 
 
 
 CO 
 
 CO 
 
 s 
 Ji 
 
 8 
 + 
 
 8 
 I 
 
 + 
 
 « i<i 
 
 ^ 
 
 s 
 
 + 
 
 -§ 
 
 8 
 
 
 -i 
 
 a 
 + 
 
 8 
 
 o 
 
 bC 
 O 
 
 1— 1 
 
 
 ^ 
 
 ^ 
 
 iHl 
 
 Ho. 
 1 
 
 1 1 
 
 II 
 
 11 
 
 
 e 
 
 
 -^ 
 
 
 ^ 
 
 '^ ^ 
 
 tH 
 
 -« 
 
 
 + 
 
 
 CO 
 
 
 CO 
 
 
 
 
 
 
 II 
 
 II 
 
 
 ^ 
 
 -i 
 
 w 
 
 -§ 
 
 -i 
 
 H 
 
 rJa 
 
 + 
 
 e 
 
 5^ 
 
 + 
 
 -§ 
 
 + 
 
 s 
 
 + 
 
 
 I e 
 
 + 
 
 a. 
 + 
 
 I 
 g 
 
 CD 
 
 CD 
 
 CO 
 
 CO 
 CO 
 
10 RATIONAL ALGEBRAIC FUNCTIONS. 
 
 C. — Expressions Involving (a -{- bx + cx^). 
 Let X = a + bx -\- cx^ and y = 4 ac — b"^, then 
 
 „„ rdx 2 ^ .2cx + b 2 ^ , .2cx-\-b 
 
 67. I ^ = —7= tan-^ ■ y^- , or — —= ■ tanli-^ — 
 
 J A -\/q -yjq V— q V — q 
 
 „_ Cdx 1 - 2cx + b — V— a 
 
 68. I — = — =log 7^' when 7 <0. 
 
 V— 9' 2cx + b + V— 2- 
 
 rdx 2cx + b 2e rdx 
 
 6 c" /^(Za; 
 
 Z' 
 
 r dx _ 2cx-{-b 2{2n-l)c C dx 
 
 -^ Cxdx 1 - ^^ & /^(/a; 
 
 72 I = — lof X I 
 
 ' • J X 2c ° 2cJ X 
 
 rxdx _ bx-\-2a b Cdx ^ 
 
 J "X^ ~ Jx ^J x' 
 
 r xdx _ 2 a + bx b (2 n — V) r dx 
 ■J A'" + i~~ nqX'' nq J X"' 
 
 „_ rx^ - a; i T ^ b^ — 2ac rdx 
 
 ^«- J 5=5''-" = ^ ^ix + tJ x' 
 
 r a;'" dx x"'-^ n — ??^ + 1 ^> /* a:"'-^rfa; 
 
 Jx" + i~ (2 71 - 7M + 1) c.Y" 2?i-m + l"cJ X'" + i 
 
 m — 1 a rx"'~^dx 
 
 "*" 2 M - w + 1 ' c J X'' + i ' 
 
*dx 
 X 
 
 RATIONAL ALGEBRAIC FUNCTIONS. 11 
 
 ^„ rdx 1 , x^ b rt 
 
 r dx __b_. ^ _ JL , /^i!_ _ i\ C— 
 
 J x^X'^2 a" ^^ x" ax'^\2a'' aJJx' 
 
 an C ^^ — 1 n + m — 1 h f* dx 
 
 J aj'^X^+i ~ ~ (?» — 1) ax"'-'X" " m — 1 a J a;"'-iA'«- 
 
 2n + m — \ c r dx 
 m — 1 aJ x"'-2X» + i 
 
 r_^ ^^ ^ C^^ 1 r dx 
 
 J ^^X'~2a(r^-l).Y«-l 2 a J X» "*" a J xX«-i' 
 
 
 /" (a' + ^>'cc)(Zx _ j; 2a'c-bb' rdx 
 
 ^^•J X» ~ 2(w-l)cX"-''^ 2c J X»' 
 
 86. r (a' + i'a:)'" X« dx = — ^ — ^— ( ^-'C^^' + b'x)"'-'X"+ 
 
 J ^ ' {in -\r 2 n At V) c\ ^ ' 
 
 + (m + n){2 a'c - 6^>') f(a' + 6'a;)'"-iX"c?a; 
 
 - (m - l)(a6'=' - aW + ca'^) ("(a' + b'xy'-^X«dx\ 
 
12 RATIONAL ALGEBRAIC FUNCTIONS. 
 
 r (a' + b'x)'"dx _ 1 f (b + 2 ex) (a' + b'xf 
 J J:» ~q{n-l)\ .Y"-i 
 
 -2(m-2n + 3)cJ ^ — -j;^^ 
 
 + m(2 a'c - bb')J ^ j;;^! J 
 
 _ 1 f b' (a' 4- ^''x)"-^ 
 
 ~ {m-2n + l)c\ X"-"^ 
 
 + (m-n) (2 a'c - i&') J ^^ ji 
 
 - (m - 1) (ab'' - aW + ca'') f ^^-^^^^^^~^\ 
 
 J (ct' + i'x)"' 
 
 1 / - VX^ 
 
 b"{m-l)\(a' + b'x)'—^ 
 
 X'^-'^dx 
 
 + n{bb'-2a'c)^— 
 
 (a' + b'x)"'-'^ 
 . „ C X—'dx \ 
 
 1 / 4- ^-'X" 
 
 
RATIONAL ALGEBRAIC FUNCTIONS. 13 
 
 89 C—^^— 
 
 J (a' + ^»'x)"'X» 
 
 b' 
 
 n v(«' 
 
 (m -1) (ab'^ - aW + ca''') \{a' + ft'a;)"— ^X"-' 
 1 / 6' 
 
 2 {ab'^ - aW + ca'^) \{n - 1) (a' + ^»'a;)"'-U'"-» 
 
 + (2 a'c - bV) (—TT-^. TF- 
 
 (m + 2n-^)b'^ r dx \ 
 
 If aJ'2 - aW + ca'^ = 0, 
 dx 
 
 J Ca' 
 
 (a' + b'xyX" 
 
 1 (- 
 
 (bb' -2a'c)\(a' 
 
 (m + 71-1) (bb' - 2 a'c) \(a' + b'xyX''-^ 
 
 + (m + 2n-2)c f—r-nr^ — pf-Y 
 '^ ^ J (a' + b'x)"'-^X"J 
 
 D. — Eational Fractions. 
 
 Every proper fraction can be represented by the general 
 form : 
 
 f(x) ^ g,x"-' + gr,x"-^ + g.x'^-' + . . ■ + ^^ 
 F(x) x" + kix"-^ + k^x"-- + • • • + A;„ 
 
 If a, b, c, etc., are the roots of the equation F(x) = 0, so 
 that 
 
 F(x) = (x- a)P {x — by (x - cy • • -, 
 
14 RATIONAL ALGEBKAIC FUNCTIONS. 
 
 then 
 
 F{x) {x-ay {x-a)P-^ (x-ay-^^ x-a 
 
 (x- by {x- by-' '^ (x- by--' '^ x -b 
 
 1 ^1 ^_ ^2 , ^3 |_ . . . 1 ^'r 
 
 (X — Cy (X — 6')*' ^ (X — C)''~^ iC — C 
 
 I ••• ••• ••• ••• ■•• 
 
 where the numerators of the separate fractions may be deter- 
 mined by the equations 
 
 •) 
 
 _, , , f(x) (x -a)' , , ^ fix) (x -by , 
 
 If a, J, c, etc., are single roots, then j? = gr =?•=••• = 1, 
 and 
 
 i''(a;) cc — a x — b x — c 
 
 The simpler fractions, into which the original fraction is 
 thus divided, may be integrated by means of the formulas : 
 
 . hdx _ r h d (inx -{- ti) _ h 
 
 r hdx _ r 
 
 ' J (mx + 71)' J 
 
 (mx + ny J m(mx + n)' m(l — l) (mx + ii)'-' 
 
 and I ■ — = — log (mx -f n). 
 
 J mx + n ni ' 
 
RATIONAL ALGEBRAIC FUNCTIONS. 15 
 
 If any of the roots of the equation f{pc) = are imaginary, 
 the parts of the integral which arise from conjugate roots 
 can be combined, and the integral brought into a real form. 
 The following formula, in which i = V — 1, is often useful in 
 combining logarithms of conjugate complex quantities : 
 
 log {x ± yi) = 1 log {x" + if) ± i tan-^ ^- 
 
 The identities given below are sometimes convenient : 
 
 1 _ 1 r h _ b' "I 
 
 {a + bx'') {a' + b'x^) ~ a'b - ab' ' \_a + bx'' a' + b'x'^J 
 7)1 + nx 1 
 
 {k + Ix) {a + bx + cx^) al^ + chP' - bkl 
 
 Vl(ml — nk) c(nk — '>nl)x + (aln + ckm — blfn)~\ 
 \_ k -{- Ix a -{- bx -{- cx"^ J 
 
 I + mx^ 
 
 {a + te") (a' + i'^") a'b 
 
 1 r^^ — am a'm — b'I~\ 
 
 — ab'' \_a + bx" a' + b'x" J " 
 
 1 ^ +JL^ + J^, 
 
 (x + a) {x + b) (x + c) X + a x + b x -\- c 
 where 
 
 B = - —Z TJ C = 
 
 {a — b){a — c) {b — c){b- a) (c - a) (c — b) 
 
 (x + a)(x + b)(x + c)(x + ;/) x + a x + b x + c x + g' 
 where 
 
 ^1 = T, : TT T ' ^ = Z 7T~/ 7T7 7^ ' ^tc. 
 
 {b — a){c-a){g — a) {a - b){c — b){g — b) 
 
16 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 III. IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 A. — Expressions Involving Va + hx. 
 
 The substitution of a new variable of integration, 
 y = Va + bx, gives 
 
 2 
 
 2 
 
 ^a^bxdx = — V(a + bxf 
 
 «« C I T- , 2 (2 a - 3 ^>a;) V(a + bxf 
 
 92. J xVa + bxdx = ^ ^g^^a ^" 
 
 «„ r , / — T^- , 2 (8 a^ - 12 abx + 15 b^x^) V(a + Z»a;)« 
 93. J xWa + Z-xc?x = -^ ^^^, '- ^ 
 
 94. I dx = 2Va + Z^x + a I — ■ 
 
 98./- 
 
 (/a; _2^a + bx 
 i + bx ^ 
 
 96. I -7= = ^^r7-„ Va + bx. 
 
 Va + bx 
 
 /xdx 2(2 a — bx) I 
 ■ I = ^ Q7.2 Va + 
 v./ 4- /ax 3 ^>2 
 
 „„ f a;2t/a; 2 (8 a^ _ 4 a^,a; + 3 ^»V) / — 
 
 97. I , = ^ .^ ,„ ^ Va + ^aj. 
 
 00 r ^^ 1 , / VaTTx — Va . „ ^ „ 
 98. I — ■ = — ^ log ( , — ], for a > 
 
 99 
 
 a; Va + bx Va \ Va + ^a; + Va/ 
 
 /c?a; 2 ^ , (a+^ —2 , , ia+bx 
 — ■ = . — tan-i \^ , or —-= • tanh-^ \ 
 
IRKATIONAL ALGEBRAIC FUNCTIONS. 17 
 
 dx Va + hx ^ C ^^ 
 
 , «« r dx ■\Ja-\-hx b r d 
 100. , = 77- I 7= 
 
 •^ x^s/a + bx «^ ^^'^ xy/a 
 
 + bx 
 
 2±n 
 
 4± n 2 ±n 
 
 102. J .(. + fa)*.& = p [_L__i L__LJ. 
 
 r a;^c?a; _ 2a;"'Vtt + to 2 ma r x'^-^dx 
 
 ' -^ Va + bx (2 w + 1) i (2 ?/i + 1) bJ Va + ^ 
 
 /* (/a; _ Va + ^>x (2?^-3)& T dx 
 
 ' '^ x"V^i + bx~ (n-l)ax"-^ {2n-2)aJ x"-Wi 
 
 105. Jfc±M*: = , J(, + fa)"-i-=rf, + a/(^ 
 
 n dx 1 /* (/a: b C dx 
 
 106. 
 
 n — 2 
 
 da;. 
 
 , /^ dx _ 1 C ___dx b^C 
 
 x{a + bxy^ x(a + ia;) 2 (a + te)2 
 
 107. ^ fix, V^^x)dx = '^fff^lL:-^, :J\z—-'dz, 
 where z" = a -\- bx. 
 
 m + n 
 
 /, , , !^ , n(a-}-bx) « 
 (a + bx) n dx = - \,,\ 
 
 /m p 
 
 f(x, (a + bx)", (a + bx^, • • ')dx 
 
 where f = a + bx, and s is the least common multiple of n, 
 q, etc. 
 
18 
 
 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 B. — Expressions Involving Both Va + hx and Va' + b'x. 
 Let u — a + bx, v = a' + b'x, and k = ab' — a'b, then 
 
 dx 
 
 k + 2bv r- ^' 
 dx = — . .,, Vmv 
 
 110. r^^ 
 
 / -^ vdx _ 1 / — h_ C 
 V^ & 2bJ ^uv 
 
 xdx V?<v ab' + a'b C dx 
 
 / 
 
 111. 
 
 112. / 
 113./ 
 
 4 bV ' "" 8 66'c; v^ 
 ^c dx 1 / — h r dx 
 
 \ uv 
 dx 
 
 bb' 2 iZ- 
 
 2 
 
 ?^/: 
 
 'wy 
 
 log(V^»^;'w + ^»V?;) 
 
 2 ^ . I- b'u 2 ^ , 
 
 = — , tan~ '■ \\—, J or — == tanh" 
 
 1 . , 2 bb'x + a'b + ab' 
 I sm- ^ 
 
 114. f- 
 
 1 , b'V^-^/kb' 
 lop- 
 
 116, 
 
 dx 
 
 V^ ~ Vkb' ^^^ 6' vw + Vkb" " ^^kb 
 
 dx 2 V« 
 
 2 , ^;'V'w 
 tan~^ — ^ 
 
 -^-kb' 
 
 /dx 
 v\uv 
 
 kV^ 
 
 116. /.."vr,.. = (2;;^,(2 .— v;; + ;./^). 
 
 rV^^ 1 / 2Vw r dx \ 
 
 (7/1 - 1) b' \ i;'"- 1 ^ ^ "J „m-l ^y 
 
 v"'-'^dx^ 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 19 
 
 /dx 1 / Vm , , ,. , r d-^ \ 
 
 120. fv^u^-idx = — —. — — r-Y 2 y'^ + iM'-i 
 
 J (2 7H -j- 2n + 1)0 \ 
 
 + (2 w - 1) k Cv^'u^-^dxy 
 
 1^ 
 
 (2n-l)k 
 
 121. Cv'"u-<'' + i^dx = — — ^ ,, , ( 2 y'« + i?t-("-J) 
 
 (2 m - 2 w + 3) b' Cv"'u-^"-^^dx j 
 
 — v"'u-^"-^^ 
 
 (2n - 1)^'' 
 + mb' Cv'"-'^u-'^"-i''dx \ 
 
 122. Cv-'^u^'^-i^dx = tA T-rA 2 ti"-^-^'"-'^ 
 
 J (2 m — 2 ?i — 1)6' \ 
 
 + {2n- l)kCu''-^^v-"'dxj 
 
 — u^ — ip—^™ — ^^ 
 
 (m- 1)^''' 
 + (*^ - i)^ r«"-it'-('"-')rfa; j. 
 
 123. ry-'"%-("+»<^a; = — -— -( 2 7-('»-i>w-("-« 
 
 J (2 w - 1) ^ V 
 
 + (2 m + 2 ?i - 3)b'Cv-"'ir <"-«c?a; j. 
 
20 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 C. — Expressions Involving ^x^^a? ani> ^a? — x^. 
 24. f Vx2 ±a?dx = \ [x Va;2 ^ci'^a? log {x + Vx^ ± a^)].* 
 
 
 X 
 
 dx . , cc .a; 
 
 — =r sip-t-, or— cos~^-' 
 
 28. 
 
 29 
 
 /c?a; 1 .a 1 , x 
 ; = - cos~^ -1 or - sec ^-■ 
 x^x^ — a? a X a a 
 
 r dx __i, A, + v.-±.^ y 
 
 -^ xVa-±a;2 « \ a; / 
 
 on C ^«^ =*= ^^ 7 /~^ ; 1 «. + Vft^ ± X^ 
 
 i*"- I rfx = Va"* dr x'' — a lo£? ■ 
 
 ^ X - X 
 
 r V.r- - a^ /— : a 
 
 I — — dx = Vx^ — a^ — a cos -• 
 
 »^ X X 
 
 32 
 
 ^ V 
 
 „„ /* xt^x /-^ 
 
 33. 1 , = Vx2 - a". 
 
 *^ Vx^ — o? 
 
 34 
 
 fxVx^ia^^^x = iV(x2±a2)S. 
 35. rxVa--x2(^x = - ^ V(a2 _ a:2)3. 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 21 
 
 136. C>/{x^±aydx 
 
 2 z 
 
 dx 
 
 dx 
 
 :X 
 
 = ^\ x-\/(x^±d 
 
 137. f-\{a^-x^ 
 
 138. f 
 
 139. /- 
 
 140. /- 
 141 I ~ 
 
 * J -».//^2 _ o,.2\3 -v/«2 _ ^a 
 
 »')]■ 
 
 V(a;2 zh ay a^ Vx^i^ 
 
 dJx X 
 
 iCf^a; — 1 
 
 xc?a; 1 
 
 142. fa; V(^2^^(ix = ^y/(x'±ay. 
 
 143. Cx^J^F^^^^dx = - |V(^2_ 3.2)6; 
 
 144. faj^ Vx=^ ± a^cZx 
 
 = ^V(x2±a^«rF|x-N/^^±^^-|log(a;+Vac*-ba?) 
 
 8 
 
 a;'e?x 
 
 145. Cx^y/^^ 
 
 log 2 = sinh - J (^^) = cosh - 1 (^^7^) ; tanh- 1 2 = - i • tan- \ziy 
 
 * (See Note ou pages 20-21.) 
 
22 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 /Va^~±x^ dx ->Ja^ ± x^ IT dx 
 
 X Lx -^ a-Va^'zhx^ 
 
 147, fx-Va^ + x^ (^x = (± 1 x2 - ^2^ a2) V(«2 i x^)^ 
 
 /^ fix Va^ ± x^ 1 /^ <^x 
 
 148 
 
 xWa-±x- ^^^ ^ la-J X 
 
 -^a? ± x" 
 
 C ffx Vx^ — a^ , 1 , /x\ 
 
 J x'^xJ' — o? 2a-x2 2^3 \^ay 
 
 160. I , = o Vx'^ ± a^ ^ - log (x + Vx^ ± a^). 
 
 151. I = — o Vct^ — X- + 77 sm !-• 
 
 , ^„ . c?x Vx^io^ 
 
 152. 
 
 J /• rfx 
 
 ^ x^Vx^ zh a.' 
 
 2 a^x 
 
 183. f- ■'^ "'""-" 
 
 x' 
 
 154 
 
 ^ = + log (x + Vx» ± a^. 
 
 X- X a 
 
 156. f / = ,--^ 4-log(x+V^^T7^. 
 
 X^(?X X . , 35 
 
 , - — = ■ , — sm-i-' 
 
 • (See Note on pages 20-«l.) 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 23 
 
 air \ du 
 
 158. Cfj^^ = ,jCf( 
 
 g'^ — cu^ j (^^ — cu^^ ' 
 
 where u — 
 
 159. I , - = - \ f( 1 du, where u^ = a -^ cx^. 
 
 J Va + ex'' cJ ' \ c J 
 
 D. — Expressions Involving Va -\- bx + cx^. 
 Let X = a -\- bx + cx^, q = 4: ac — b^, and k = In order 
 
 q 
 
 to rationalize the functio n f(x, Vft + bx + cx^) we may put 
 Va + ^x + ex' = V± c V^ + Bx ± x^, according as c is posi- 
 tive or negative, and then substitute for x a new variable z, 
 such that 
 
 z = ^A + Bx + a;2 ± a-, if c > 0. 
 
 V^ + Bx — X' — vCi a ^ 
 
 z = ■} if c < and >• 0. 
 
 X — c 
 
 = \- -f where a and /? are the roots of the equation 
 
 ^ a — X 
 
 a 
 
 A -\- Bx - x^ = 0, if c < and < 
 
 — c 
 
 By rationalization, or by the aid of reduction formulas, may 
 be obtained the values of the following integrals : 
 
 160. C^ = ^-logfVX-\-xVc + -^\iic>0. 
 ^ -\fx Vc \ 2Vcy 
 
 /dx —1 /2c.r + i>\ 1 . , ,/2c.>' + h\ 
 
24 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 ' -^ X^fX q^/X 
 
 r dx _ 2(2cx + b)Vx 2k {71- 1) r dx 
 ^^" J X" VX ~ (2 n - 1) ^A'« "•" 2 w - 1 J X«-i Vx' 
 
 67. Cx^VXdx 
 
 ^ {2cx + b)^X f hX , 15\ 5 r dx 
 
 12c \ '^ 4k Sk'J Wk^J VX 
 
 68. rx-vx..^ (^-+^)f:^^ + ,;"V, r^. 
 
 J 4(7i + l)c 2(% + l)A;J Vx 
 
 /xdx^_ Vx b r dx 
 VX~ c '2 c J Vx 
 
 dx 
 
 Vx" 
 
 69 
 
 /* a;c?cc _ 2 (^ij; + 2 «) 
 
 ■ J xVx ~ ^Vx 
 
 / xdx Vx A r ^^ 
 X"VX~ (2ri- l)oX«~2"J J^VX' 
 
 "• J vf = (ro - jp) ^ + -87- J vx • 
 
 ' J xVx c^Vx c J Vx' 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 25 
 
 174 
 
 Vn 
 
 x^dx 
 
 x^Vx 
 
 175 
 
 {2b''-4:ac)x + 2ab 4 ac + (2 n - 3 ) 5' P dx 
 {2n-l)cqX'^-^Vx (2n-l)cq J X^-^Vx 
 
 x^dx 
 
 ■f 
 
 Vx 
 
 'x^ 5bx 5b^ 2a' 
 3 c 12 c2 "^ 8 c3 3 c2 
 
 X + 
 
 '3 ab 5 b^ 
 4 c^ 16 & 
 
 )f 
 
 dx 
 
 Vx' 
 
 176. fxVXdx = ^^^^~~ f-yXdx. 
 J 3 c 2 cJ 
 
 177. fxXVxdx = ^V-^ - ^ fxVx dx. 
 J 5 c 2 cJ 
 
 xX^dx X'^VX 
 
 P 
 
 S 
 
 X^dx 
 
 Vx (2*i + l)c 2cJ Vx 
 179. Cx^VXdx =(x-^}j 
 
 180 
 
 5 b\ X^X 5P-4ac 
 
 4:C 
 
 16 c2 
 
 /Vx 
 
 c?a;. 
 
 r x'^X'^dx _ xX'^Vx _ (2w + 3)^' r xX'^dx 
 ' J Vx ~2(?i4-l)c 4(7i4-l)cJ Vx 
 
 ^— r 
 
 2 (71 + 1) c J 
 
 181. Cx^Vxdx = (x^- 
 
 X^lx 
 
 (n + l)cJ Vx" 
 
 8 c 48 c' 3cy 5 c 
 '3ab 
 
 + 1 
 
 8c2 
 
 32cvJ 
 
 VXrf; 
 
 'JCt 
 
 -^ xVX Va \ a; 2Va/ 
 
26 
 
 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 183. I — = — ^sm-M -.= ] , or — = smh ^ ^• 
 
 184. f ^ 
 
 dx 
 
 2VX 
 
 — , ' II a = 0. 
 
 ox 
 
 185. 
 
 d 
 
 xX" 
 
 Vx 
 
 Vx 
 
 (2 n - 1) aX 
 
 1 r dx b_r 
 
 " aJ -rX^'-'^^fx 2aJ 
 
 dx 
 
 186 
 
 ■/ 
 
 dx 
 
 Vx 
 
 a;'VX 
 
 
 dx 
 
 x«Vx 
 
 «« 2aJ a;VX 
 
 188 
 
 »^ X 
 
 X 
 
 189. fj^^iKl^ 
 
 Vx (2 ?i - 1) Vx 
 
 Vx 
 
 '/ 
 
 X"-''dx . ^> rx^'-^dx 
 
 X 
 
 Vx 
 
 iC^ 
 
 a? 
 
 + « I ^ + 
 
 dx 
 
 .xVX -' Vx 
 
 U-- 
 
 VJ 
 
 2-' xVx ^ 
 
 dx 
 
 190 
 
 Vx' 
 
 191, 
 
 r x'^dx _ 1 r x'^-^d,x _ h_ C x-^-^dx a Cx'^' 
 ' ^ x»Vx c J X'^-WX cJ A'«VX cJ x» 
 
 f 
 
 ■-r"'X"6ga; _ x"'~'^X"Vx _ (2n + 2m-l)h P x^^-'^X^dx 
 Vx ~ {2n + m)c 2c(2n + m) J VX 
 
 {vi -\)a r x'^-^X'^dx 
 {2n + 7n)cJ Vx 
 
 192, 
 
 r dx 
 
 Vx 
 
 (2n+27n-3)l> 
 
 (w — 1 ) ax'" - ^X " 2(t{m~l) 
 
 ?i + m — 
 {in — 1) « 
 
 '> r dx_ 
 
 'Vx 
 
 (2 ?^ + m - 2) c Z' 
 
 rf.r 
 
 x 
 
 m-2^Y«VX' 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 27 
 
 ■ 
 
 r X"dx _ _ x"-'Vx (2?i-i)^> r X"-'(ir 
 
 (2 7^ - 1) r X''~^dx 
 m. - 1 J ^-'"--'VX 
 
 194. r/(a-, V(a; - a) (x - ^')) fZx 
 
 /\hu^ — a u (b — a) [ u du 
 
 where u^(x —b) = x — a. 
 
 E. — Expressions Involving Products o f Powers of 
 
 (a' + b'x) AND Va + bx + cx\ 
 
 Let X = a + bx + cx^, v = a' + b'x, q = 4:ac — b^, 
 /3=-bb' -2 a'c, k = ab'^ - a'bb' + c«'^ then 
 
 195. I — — = —j=log "^ — 
 
 yVX VA; 
 
 tan"^ 
 
 V- k 2V^-kX 
 
 = sin~^ 7=="' iiA;p£0. 
 
 V— ^ b'v^—q 
 
 „ r dx 2 b' Vx . „ , „ 
 
 196. — p= = 7. ' if ^ = : 
 
 ^ v^X 
 
 (So 
 
 r dx _ ,_ ■> /^ + 1 , 
 
 /- _^ ^ _ &'Vx _ _^ r dx 
 
 ■J r^Vx" ^'^ 2aJ vVx 
 
 ■Jy^VX 3/3i;^ 3/3J ^;VX 
 
28 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 200 f ^dx ^ 2(2k+Jv) 
 
 ««, rvdx b'^^ B r dx 
 
 201. -7= = TT I -1^' 
 
 202. Cv^Xdx = ^-^^^-i~^Vxdx. 
 
 r vdx __ _ b'Vx _ ^ r dx 
 J X" Vr " ~ (2 .. - 1) cX» 2 J A'« VX ■ 
 
 r^^x^ _ &'x»Vx _ _^ rxv^ 
 
 J VX ~ {2n + l)c~ 2cJ VX ' 
 
 r (Za: _ _ bWx _ (2m-3)f3 r dx 
 ■ J ^'«VX (m — 1) A-y™-' 2(m — 1)A-J ^,m- iVX 
 
 _ (m-2)c p dx _ ^ ifyt-z£0. 
 (m — l)A-c/ ym-aVx 
 
 206 
 
 1/ y; 
 
 ,y-«Vx (2 m -1)^^;" 
 
 _ 2(m-l)c f _^^_ ■ . , _ rt 
 
 {2m-l)(^J ,.»-iVx' 
 
 r Vx^x _ _ ^>'XVX _ (2m-5)/3 T Vx^ 
 J t;-" " (m-l)A;v"'-i 2 (m - 1) ^- J t-""-! 
 
 (m - 4) c rVxdx 
 ~ (m- l)kJ v"'-^ 
 
 ~(w-l)^*'2(^ *'"'-■ ^^J i;'"-'Vx"^ ''J i;"'-2Vx^ 
 _ 1 /_ &'VX _ r dx _ J r rfa; \ 
 
 ~ (m - 2)6'='(^ v—' V v'-Vx ^^J v-'-'VxJ 
 
IRRATIONAL ALGEBRAIC FUNCTIONS. 29 
 
 208. (v'-<Xdx=- — \-—-{vv^-^X^'x 
 J {in-\- L)c\ 
 
 - {m+ ^) ^C v'"-'^^ dx- (m-l)kC v"'-'-Vx dx\ 
 
 ^^« f* dx 
 
 209. 1= 
 
 1 / h'Vx , , , ^^^ c dx 
 
 + (m + 2n-2)c f ^^ ^ Y if A;5z£0. 
 
 ■ J i'- .Y« VX ~ {2m + 2n-l)li\ v"' X» 
 
 + (m + 2?i-l)c f ^^-T=Y if A; = 0. 
 
 * r X"dT 
 
 211. I ^L^^, 
 
 1= 
 
 v^'-WxJ 
 
 + (2n-l)kf- 
 ,, ^ r X''-^dx\ 
 
 
 A'»-ic?a; 
 
 1 / &'X"-Vx ..^f x»- 
 
30 IRRATIONAL ALGEBRAIC FUNCTIONS. 
 
 213 
 
 Vx {m-\-2n)c\ 
 
 v"" (Zx 1 fVv^-^ Vx 
 
 'v^'-^X'^dx' 
 
 /V" 
 Xn 
 
 Vx (m — 2 ?i) c \ A'" 
 
 ^ '^^J X«VX ^ ^ -^ X"VaJ 
 
 
 + 
 
 {x + a) (x + Z*) VX (S - a){x + a) Vx (a — b)(x + b)^X 
 
 1 
 
 Va + bx -\- cx^ ± Va' + b'x + r'j:- 
 
 _ Va + 6a; + (?a;^ ip Va' + b'x + c'. 
 a — a' + ((^ — ^') a; + (c — c') x^ 
 
 Vx Vx Vx 
 
 a;2 
 
 (x + a)(a; + 6) (^' — a)(a; + a) (a — Z») (cc + J) 
 
 (a- + a)VX _^rf^^ (a-Z^)Vx 
 X + 6 x + b 
 
 C lax' + Z* _ . „. . . 
 
 c/ \ ' -2 j_ i i^^ IS ^^ elliptic integral. 
 
 /cc Va + bx^ , 1 r , 
 ; dx = — I Wab' — a'b + by^ ■ dy, 
 Va' + b'x"" bWb'^ ^ ^ ^> 
 
 ' x Va + ^a;'-^ 
 where y^ = a' + b'x\ 
 
MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 31 
 
 IV. MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 
 
 V2 ax — x^ •dx = — ^ — ^sr2ax — x^ -[- — sin"^ 
 
 a 
 
 «,^ r dx . . X 1 /-, ^\ 
 215. I — - = versm"^ - = cos~^( 1 ) 
 
 ^ V2 ax - x2 ct \ */ 
 
 ^-« . a;"c?a; a,w-iV2acc — x^ 
 
 216. 
 
 V2aa;- 
 
 ^2 71 
 
 a (I -2 n) C x^'-'^dx 
 
 / 
 
 n J V2 aa; - x^ 
 
 / 
 
 217 
 
 *^ a;''V2 ax — aj^ 
 
 + 
 
 dx a/2 ax — x^ 
 
 x»V2ax-x2 a{\-2n)x- 
 
 n — 1 C dx 
 
 1) a J rpU—l 
 
 (2 w - 1) aJ a;«-i V2 ax - x^ 
 
 //- 5 , x"-iV(2ax — x^)" 
 X" V2 ax -x^-dx= ^^-— r ^ 
 71 + 2 
 
 _^ (2.^ + l)a r„_,V2^^3T.., 
 w + 2 J 
 
 r V2 ax - x^ • 6?x _ V(2 ax - x'')^ 
 
 71 — o /' V2 ax — x^- c?x 
 
 — 3 /' V2ax -c 
 (2 71 - 3) a J x»^ 
 
 220. I — , = — sec-^ ( - )• 
 
32 MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 
 
 221 
 
 •/ 
 
 dx 1 , Va^ + x^ — a 
 
 , = — log , 
 
 a;Vx"-|-a2 an Va^ + a;" + a 
 
 222 
 
 223 
 
 /x?dx 
 
 %■ sin~l / — 
 
 Sin 
 
 CC"' 
 
 a 
 
 / 
 
 • (Za; 
 
 (a + ^'a;2)Vx hV"^ 
 
 log I 
 
 'a; + 8' + V28^' 
 
 + tan-M 1 + 
 
 '2x 
 
 -tan-M 1 
 
 \ 
 
 '2x 
 
 where iS* = a. 
 
 c^.dx 1 r^ ,/^ , V2x\ ^ y. V2.t\ 
 
 a + 6ic^ 
 
 — log ( , — ) )■ , where hV' ^ f- 
 
 V a + bx? 
 
 225. 
 
 ^ -dx 2 V« 
 
 / x^ -dx _ 2Va; a T 
 
 c?a; 
 
 (a + hx?) Vx 
 
 226 
 
 c?a; 
 
 Vac 
 
 + 
 
 r 
 
 dx 
 
 227. 
 
 ■-^ (a + ^-x2)=2V^ 2a(a + Z»a;2) ' 4 a J (« +7»a;2) V^ 
 /^ ^sfx-dx x^ . 1 r ^Jx-dx 
 
 + 
 
 (a 4- ^'a;'-')^ 2. a{a -\- hx^ 4aJ (a + bx^) 
 
 iaJ (a 
 
 If tti, ttj? «8> etc., are the roots of the equation 
 
 Pffic^ +^ia;"-i +x>2^''~'^ + • • • +i9„ = 0. 
 the integrand in the expression 
 
 / 
 
 {P(fic" + piX"~^ + • • • +p„) Va + 6x + cx^ 
 
MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 33 
 
 where m<in, may be expressed as the sum of a number of 
 
 partial fractions of the form , ^==i and these 
 
 {x — af?f^ a -\-hx-\r cx^ 
 
 can be integrated by the aid of equations given above. Thus, 
 
 228. f (P- + 9)dx 
 
 J (x- a') (x - b') Va 4 
 
 + bx -\- cj? 
 
 _q -\- a'p P dx 
 
 ~ a'-b' J (x-a')Va 
 
 229 
 
 ■ f- 
 
 -' (a' 
 
 (x — a') Vo. -\- bx + cx^ 
 
 q + b'p f* dx 
 
 a'-b' J (x - b') -^a-\-bx + cx^ 
 
 dx 
 
 (a' + c'x") VoT 
 
 cx^ 
 
 ^ f an-1 ^ ^ («C' - «'«) 
 
 tan""^ X 
 
 -\/a'{ac' — a'c) ^ a' (a + cx"^ 
 
 1 , Va '( a 4- ea-- ) + .r Va 'c — nr' 
 log ^ 
 
 230 
 
 2 Va'(a'c — ac') ^a'{a + ^a'"^)- x Va'c — ac' 
 (a' + c'a;^) Va + cx^ 
 
 1 .,..-, |o'fa + ex') 
 
 Vc' (a'c — ac') ^ a'c — ac' 
 
 1 1 Vc' (a 4- ex*) — Vac' — a'c 
 
 2Vc'(ac'-a'c) " Vc'(a + cx^) + Vac' - a"c 
 
 231. \fU, \/^n^[^^^ 
 
 where z"(a' + 6'x) = a -\- bx. 
 
 z"~'^dz 
 
34 MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 
 
 232. ffi^, V c + V'a + bx) dx 
 
 where «" = c + Va + bx. 
 
 where y* (a' + b'x) = a -\- bx and s is the least common multi- 
 ple of n, q, etc. 
 
 234. ff(x, ■Va + bx + a;2) dx 
 
 r f2Va-z-b s^Va-bz+Va \ (z^Va - bz +Va)dz 
 
 where xz + Va = Va -\- bx + x\ 
 
 235. ("/(a;, Va + ^»cc + x"^) dx 
 
 - C A '^^~^ V? - bu -V aVl{bu - a - ti^) du 
 ~J'^\b-2u' 2u-b J {b-2uf ' 
 
 where u = Va + bx + x^ 
 
 x. 
 
 'x2+axV2 + aA „^ _,/aa;V2' 
 
 -'a;^ + a* 4a8V2l W-ax V2+W 
 r_rf^ = J_ f lo, f ^::^«A _ 2 tan- Y^^ I 
 
 a^— x^. 
 
TRANSCENDENTAL FUNCTIONS. 35 
 
 V. TRANSCENDENTAL FUNCTIONS. 
 
 236. j sin x -/(cos x)dx = — \ /(cos x) d cos x. 
 
 237. j cos X -/(sin x) (/a; = | /(sin a-) rZ sin a;. 
 
 238. rsinx-/(siua;, cos x)dx-^- \f{^^ - «^ «)<'^> 
 where z — cos x. 
 
 239 r__^^-_ = _l— / f-^-f-^l, 
 *^''- J a^h cos a; e(^ - (') \J z ■\- c J z-c j 
 
 where s = tan ^x, and c- = (^^ + «)/(^ - a). [See 651.] 
 
 , , . = 1 , ^; "^, 5' where s; = tan i a;. 
 
 (X ± (^ sm a; »/ a ± 2 fts; + az' 
 
 241. ^/(sin a-) 6?x = -ffUos (^| - ^) ) '^ (^| " ^J' 
 
 242. r/(tan a-) rfa; = - J /ctn f | - xj d (^| " ^ j' 
 
 243. j"/(sec a;) dx = - J/csc f | - a- J fZ (^| ^ ^ j' 
 
 rsmx^f(sui^x)dx^ r J^z) dz 
 
 J Vl-A;^sin2x ^ 2 V(r-^^) (1 - k^^z) 
 where z = sin'^a;. 
 
 r cos.r./(cos^x).7.r ^ r.A l-^)^^ . ^here z = sin^x. 
 J Vl-A-^sin^a; -^ 2 V^ (1 - A;^ «) 
 
36 TRANSCENDENTAL FUNCTIONS. 
 
 /' tan X -/(tan^ x) dx _ C J z \ dz^ 
 
 where 2: = sin^ aj. 
 
 247. r/(«x + ^)<^^ = ^ ^ f{ax + S)c?(aa^ + ^'). 
 
 248. rsec" + 2a; ./(tan cr) (Za; = (^(l + z^yf{z) dz; « = tan 3 
 
 249. I /(sin x, cos x) dx 
 
 = — I /( cos I — — X], ?,\n\ — — x]]d\ — — x 
 
 |_^Uin(|-.))c/f| 
 
 //• <i (x) dx 
 fix) ■ sin-i x-dx = sin-^ x ■ ^ (a;) — I ^^ ^ _^ > t/ic, 
 
 where <f>(x) = i f(x)dx. 
 
 /C <b (x^ dx 
 f(x)-cos-^xdx = cos-^x-<f,(x)+j y ' ^ - 
 
 252. \f{x) ■ t2,n-^xdx = tan-^ x-<f,(x) - j t^ f ' 
 
 253. r/(a;) • ctn-^ ic dx = ctn-^ a; • <^ (a;) + f^M^ . 
 
 254. j /(a;, cos x)dx = — I ./"( o ~ «> sin z j c?2:, 
 where 2; = — — x. 
 
 Li 
 
 /' sin a; -/(cos x)dx _ IT J z — a\ dz 
 J a-\-bcosx bJ \ b J z 
 
 where z = a + b cos x. 
 
TRANSCENDENTAL FUNCTIONS. 87 
 
 256. ff(x, ^ogx)dx^jf(6', z)e'dz, where « = log x. 
 257_ r fQogx)dx ^ Cf{£^dz, where z = log x. 
 
 258. Cx"'f(log x) dx = Je('" + i>y(s) dz. 
 
 259. (/(sinar, cos a;, tan x, etna;, sec a:, csca;)c?« 
 
 ' -J-^Vr+T^' 1 + ^^' 1-^=^' Iz'l-z^' 2z ) 
 
 2dz , ^ ^ 
 
 ■ ) where z = tan - ; 
 
 1 + z' 2 
 
 VT^^2 1 1' 
 
 
 :» where z = sinaj; 
 
 Vi - «' 
 
 = I / , > z, -y VI + s^ 
 
 : J where z = tan a; : 
 
 l-\-z^ 
 
 ://(vi, vi^, Vr^^, W' vfe' 7: 
 
 :> where « = sm-'a;: 
 
 2^z(l-z) 
 
 J''\^i + z vi + v^ ^ ^ y 
 
 
 2V2(1 +«) 
 
 :j where z = tan^x. 
 
38 TRANSCENDENTAL FUNCTIONS. 
 
 260. I s\nxdx= — cos x. [See 247.] 
 
 261. I sin^ a; c?x = — ^ cos a; sin a; + 1^ cr = ^ a:; — :|^ sin 2 a;. 
 
 262. I sin^ xdx = — -^ cos x (sin^ x + 2). 
 
 «o« r • „ 7 sin"-^a; cosa; . n — 1 f . „ „ . 
 
 263. I sm''xdx = 1 I sin«-^a;c?a;. 
 
 J n n J 
 
 264. I cos a3 c?x = sin X. [See 247.] 
 
 265. I cos^ xdx = ^ sin a; cos x -'c \x = ^x + \ sin 2 aj. 
 
 266. I cos^ a; c?a; = -J sin a: (cos^ x + 2). 
 
 267. I cos" a; c?a; = - cos" ~^ a; sin a; H | cos"~^a;c?a;. 
 
 J n n J 
 
 268. I sin x cos xdx = \ sin^aj. 
 
 269. I sin^ x cos'' a; c?a; = — ^ (:|^ sin 4 a; — x). 
 
 270. I sin X cos"» a; c?a; = — 
 
 cos'^ + ^as 
 
 i.p 
 
 271. I sin"'a; cos a;c?a; = 
 
 in + 1 
 sin"» + ^a; 
 
 m+ 1 
 272. j cos"" a; sin"x(Za; = 
 
 '•/ 
 
 273. I cos"'x sin"a;rfx = 
 
 cos'^^^a; sin^ + ^a; 
 
 rth -\- n 
 m — 1 
 m + w. 
 
 sin"~^a; cos'^+^a: 
 
 4- — — r— I cos'""'' a; sin" a; (/a;. 
 
 m + w. 
 
 + "^ . ^ I cos'"a; sin"-''a:rfx. 
 
TRANSCENDENTAL FUNCTIONS. 39 
 
 , ri^^m^xdx 1 / sin"-\'r , ^ ^ rsiii"-^a-rf.»'\ 
 
 274. I ■ = \ — \-(n — 1) I 
 
 J cos™ a; n — m \ cos'"~*a; ^ J cos"* a; / 
 
 1 /siu^ + ^-c , , _, rsin"a7(fe\ 
 
 = 7 \ hi — m + Z) I r— 
 
 m — 1 \cos'"-^a; ^ ^J cos"-'*x/ 
 
 _ 1 / sin"-^a; rsin«-2xrfx\ 
 
 ~ m — 1 \cos"'^^a; ^'^ ^J gos"'-^xJ' 
 
 rcos,^xdx _ cos^' + ^a- m — ?i + 2 /'cos'"a:;c?cc 
 
 "J siii"a: (/i — 1) sin""-'^ n — \ J sin"~^a; 
 
 '^x m— 1 /^cos'""'^.' 
 
 )in"^^a; m — ?^c/ sin": 
 
 J_ r cos'"-^xd. 
 n — lsm"~^x n — lJ sin"~^a; 
 
 (m — n) sin" ^x m — nJ sin" a; 
 
 1 cos'"~^a; m — 1 /*cos"'~^xdx 
 
 . . , _cos™ [tz— x]d[— — x 
 
 , sm^xdx r V2 
 
 /sin'«a:;c?a;_ r 
 cos"ic »/ 
 
 sin" ( 77 — a^ 
 
 277. I -^ = log tan x. 
 
 J sin x cos x 
 
 278. ^i- = log tan - + - ) 
 
 J cos a; sin^a; \4 2/ 
 
 — CSC X. 
 
 279. j 
 
 sin^'x cos" a; 
 1 1 ,m + w — 2/* dx 
 
 m-\-n — 2 r _ 
 
 3"~^x n — 1 J sii 
 
 n — 1 sin"*" ^ a; -cos"" ^x n — 1 J sin"'a; -cos^^^a; 
 
 1 1 m + n — 2 r dx 
 
 ~ m — 1 sin"*~'a; •cos"~^x m — 1 J sin"*"~^a; -cos".-?; 
 
 ^ r dx 1 cos a; , m — 2 /* dx 
 
 280. I -. = r -: \ T I • ^ , • 
 
40 TRANSCENDENTAL FUNCTIONS. 
 
 dx 1 sin X . n — 2 C dx 
 
 C dx 1 sm a- ii — 2 r dx 
 
 J cos"a; n — 1 cos'^^-'a; n — \J cos"~^ 
 
 282. I tan xdx = — log cos x. [See 247.] 
 
 283. I iaxi^xdx = tan x — x. 
 
 284. I tan"a;fZic = r^ — j tan^-^xc^ic. 
 
 285. I ctn xdx — log sin x. [See 247.] 
 
 286. I ctn^xdx = — etna; — x. 
 
 287. jctn^aic^a; = - ^ "_ J^ — jctn"-^xdx. 
 
 o.^^ /^ T -1 i /''■ , »'\ ,1 1+sina; 
 
 288. I sec xdx = log tan I - + - 1 = |- log -t— 
 
 289. I sec^a;c?a; = tan x. 
 
 sin a: 
 
 /r dx sma- n — 2C «^ 
 sec^a^rfx = I = ■; TT \ — I I -^^ 
 c/ cos"a; (72, — 1) cos" ~^ a; w — l^cos"~^a; 
 
 sin a; , n — 2 C , , 
 
 = 7 ;^ r~l 1 7 I sec"~^a;ria;. 
 
 (n — 1) cos"~^a: n — lJ 
 
 291. I esc a; cifx = log tan -^ X. 
 
 292. I csc'^xc^a; = — ctn x. 
 
TRANSCENDENTAL FUNCTIONS. 4-1 
 
 293. I csc"a;rfx = | -r-^ 
 
 cos X , n — 2 r dx 
 
 n — 2r dx 
 
 "-^a; n — 1 J sin"-^i 
 
 (n — l)sm"~^x n — lJ sin"~"^ic 
 
 cos x n — 2 f „ , 
 
 = — 7 ..s . „ 1 1 7 I csc^-^xdx. 
 
 (n — 1) sm"~^a; n — lJ 
 
 294. ftr-r^^ = - tan (i tt - i x). [See 241.1 
 
 J 1+ sill X V4 -J / L J 
 
 /dx 
 : = ctn a TT - ^ x) = tan (i tt + ^ a;). 
 1 — sin a; ^ 
 
 296. 1 :; = tan A x, or esc x — ctn cc. 
 
 »/ 1 + cos X 
 
 /dx 
 = — ctn 4 a-, or — ctn x — esc x, 
 1 — cos X 
 
 298. f— -^^ = ^-^^ • tan- > (sec ^ • tan i a- ± tan 0), 
 
 if a > i, and b = a sin 6. 
 
 ^„„ /* c?.x ± sec a , sin h(a ±x') 
 
 299. I r-^ = 7 — log f-) (» 
 
 J a ± sm a^ o cos -^(x ^ a) 
 
 if 6 > a, and a = 6 sin a. [See 241.] 
 
 ««« /* dx — 1 . , r^ + a cos a;~l 
 
 300. I z = , • sm-' — — 
 
 J a -\- b cos X Va^ — b^ \a-\-b cos a; J 
 
 5 
 
 CO ? 
 
 " 5 
 
 o s 
 -Sim 
 
 QJ o o 
 
 •^ 5- 
 
 1 . , r Va'^ — b^ ■ sin a;~| 
 
 , sm- — : 
 
 I Q^ — ip- \_ a -\- b cos X J 
 
 b g or : sm- ' I r^ I ■> 
 
 or , 
 
 _ii 1 - ,rVft2-^ 
 
 £s-§ or 
 
 c o 
 
 « •a 
 
 Va* - y 
 
 tan~M y^ tan \ x V> 
 
 , r Vft^— ^ • sin a;~| 
 
 tan-M — T— h 
 
 L c» + a cos X J 
 
42 TRANSCENDENTAL FUNCTIONS. 
 
 1 , r ^* + <^ COS a; + V6^ — d^ ■ sin x ~\ 
 ' >2 - a2 °^ L a + 6 cos a; J ' 
 
 V^ 
 
 or 
 
 1 , r V^ + ft + V6 — ft • tan ^ a; ~| 
 
 '-ft' LVZ* + ft-V6-a-tanlxJ' 
 
 1 , , , r V^»2 - ft2 . 
 or 
 
 , tanh-T \^^-"^-^^^^ l 
 
 V^2 _ ^2 ^ 6 _|_ (J cos aj J 
 
 _-, r dx 1 ^, , 
 
 301. I — TTT = o , ,i, [b log (ft cos cc + ^> sm x) + ftxl 
 
 J a + 6 tan x a^ + ¥^ ^ /'J 
 
 /f/a; 1 
 
 ^ \ = -7= log tan (i cc + i tt). 
 
 r _^mxdx_ _ _ /* COS {\ TT — X) d (^ TT — x) 
 
 ' J a -\- b cos x J a + b sin (^ tt — a;) 
 
 = — - log (a + b cos a;). 
 
 304 
 
 /(ft' + J' cos x) dx _ b'x a'b — ah' r 
 a + b cos X b b J i 
 
 305 
 
 / 
 
 + b cos X b b J a-\- b cos x 
 
 (ft' + b' cos a;) c?a; _ ab' — a'b sin x 
 (a + b cos x)^ a^ — h^ a + b cos a; 
 
 ftft' — bb' C dx 
 
 , ftft' - bb' C dx _„ „,.^ 
 
 "• 2 TT I — TT [See 241.] 
 
 ft'' — 6^ c/ ft + ^ cos a; ■- -" 
 
 gQg r (ft' + ^''cosa;)c?x ^ 1 V jab' -a'b)^ix\x 
 
 ■J (ft + ^ cos a;)" (w- 1) (ft2- Z*^) |_(a + />cosa-)"-i 
 
 r r(r^^<' - ^>Z'') (» - 1) + (n - 2) (ft/>' - a'b) cos a-] r/x l 
 J (ft + 6cosa;)"-i J" 
 
307./ 
 
 TRANSCENDENTAL FUNCTIONS. 43 
 
 (a' + b' COS x) dx _ (a' — h') tan ^ x 
 
 (1 + cos x)» ~ (2 w - 1) (1 + cos x)"-^ 
 
 n (a' + h') - g.' r dx 
 
 2n-l J (1+ cos x)"-i ' 
 
 308 C——^^— = - r - ^ sin a; 
 
 J (a + b cos xf (n - 1) (a^ - ^^^^ [_(a + Z» cos a;)"-i 
 
 + (2n-3)aj ^^_^^ 'cos x)— ~ ^'' ~^U (a + b cos a;)'-^ J 
 
 309 f ^^ ^ tanja; 
 
 * J (1 + cos x)" (2 ?i - 1) (1 + cos x)"-^ 
 
 ^27i-lJ (l + cosx)»-^ L'e^-'-i^-J 
 
 otn r (*' + ^' COS x) f/a; a'b — ab' , , j . 
 
 310. I ^ — — — '- — - = — ^ -^ log (a-\-b cos a^) 
 
 »/ sin x(a + b cos x) a^ — ¥ ^ ^ 
 
 H — ^ log sin ^x log cos -A- x. 
 
 a + b ^ ^ a — b ^ ^ 
 
 »■... r ((t^' + b' cos x) dx a' ^ , ,, , . 
 
 311. I -^^ — —7-^ — - = -logtan|-a7r + a;) 
 
 J cos x(a + b cos x) a ^ ^ 
 
 (ab'-a'b) f dx 
 
 - a'b) r 
 a J 
 
 a -\- b cos X 
 
 «,« r (a' -\- b' cos x) dx 4- («' =F ^') , i / i 7im 
 
 312. \ ,., ^ = ± ,\ ^ +i(a'db&')logtan|; 
 
 J sm a; ^1 ± cos a;) 1 ± cos a; '' ^ '^ * ^ 
 
 313 
 
 /dx _ — ctn I" X 
 
 (1 - cos xy ~ (2 71 - 1) (1 - COS a;) 
 
 n — 1 C dx 
 
 n — l 
 
 4.iL:Ll.r_ ^h-—. ^See241.] 
 
44 TRAMSCENDENTAL FUNCTIONS. 
 
 dx C dx 
 
 314 
 
 r dx r 
 ci^ — b"^ cos?x J (a^ — U^ 
 
 1 . sin (a — x) 
 
 log ^^ : 
 
 2 ab sin a sin (a -\- x)' 
 
 ~l "■ — 3"t^^~M~^^ )' whei 
 a^ sm /? \sm pj 
 
 1 a 
 
 or -s -: — — tan ' i -: — - j , wnere cos a 
 
 cos /3 b 
 
 „,^ /* tZcc 1 ,/&tanic 
 315. I p^ 5 . ,„ . „ =— tan-^i 
 
 a^ cos^ a; + ^^ sin^ a; ab 
 
 a 
 
 316. r44^ = ^^ tan- ftan . . J-^") 
 J « + b co&^x b^a \ ^ a + bj 
 
 X 
 
 ~b 
 
 <«<» C sm a; cos a:" f/a* 1 , , , , • o s 
 
 J a cos- a; + 6 sura; 2(b — a\ ^ ' 
 
 318. f ^ = C d{x-a) 
 
 J {a -\-b cos X + c sin a.')" J [a + ?• cos (a- — a)]" 
 
 where 6 = r cos a and c = r sin a. 
 
 319. I : [See page 61.] 
 
 J C ' 
 
 dx 
 
 a -\- b cos X -\- c sin x 
 
 -1 
 
 sin" 
 
 J r i^ + c^ 4- a (b cos a; + c sin x) 
 ^a^ — W—c^ |_ V(&^ + G^){a + ^ cos a; + c sin a;) 
 
 , ^= • log 
 
 V&2 + e^ _ ^2 
 
 ] 
 
 [^^ + 0^4- a (& cos a- + csin a;) + V&^ + c^— ^^(isinx— ccosa;)"! 
 V(6^ + c^) (a + i cos a: + c sin a;) J 
 
 1 , V^-2 4- ,.2 - a2 _ c _j_ /^ _ (j) tan ^a; 
 
 OP 
 
 o 
 
 ' X 
 
 ■sIV + c- -a" ■\/b'' + c'-d' + c-(b- a) tan \ 
 
 c^ [_ Va^ -b^-c^ J" 
 
 Va^ - 62 
 
TRANSCENDENTAL FUNCTIONS. 45 
 
 /dx 1 
 
 — -— — : = - log (a -h c tan |a;) 
 a (1 + cos x)-\- c %\nx 'C 
 
 321 f ^^ 
 
 J (a [1 + COS a;] + c sin x)^ 
 
 1 r c (a sin a; — c cos x) i / , . i x ~l 
 
 = -o —^ r-;' ■ a log (a + c tan ix) • 
 
 ^r.r. r (a; + sin a?) c^cc ^ , 
 
 322. I ^— r^ — = a;tan-^a:. 
 
 J 1 + cos x 
 
 = |- sin a; Vl — P sin^a; + — sin~^(A; sin a;). 
 
 324. I sin x Vl — A;^ sin^ a; t?a; 
 
 = — -|- cos a; Vl — A;^ sin- x ^-j— log (k cos a; + Vl — A;^ sin^a;), 
 
 325. J sin a: (1 - A;^ sin^ x)^ dx = - i cos a; (1 - A;^ sin^ x)^ 
 
 + f (1 - k^) jsin X Vl - k"" siu^x dx. 
 
 ««„ /^ cos a;c?a; 1 . , ,, . . 
 
 326. I , . ^= = T sin-i (A; sin x), 
 ^ Vl — k^ sin^x ^ 
 
 or - log (6 sin x + Vl + 6^sin^a;), where b^ = — k^ 
 
 327. I- = — -log(A:cosx + Vl — A-^sin^a;), 
 •^ Vl — k^ sin'^ a; ^ 
 
 1 . , Jcosx „ 2 
 
 or — TSin~' , , > where 6. = — k^ 
 b Vl+62 
 
 ^„„ /• tan a; c?a; 
 
 328. - = 
 
 ^ Vl — A;^sin^a; 
 
 1 , / Vl - A;=^ sin^a; + Vl - k''\ 
 ;log 
 
 2Vi^^ \ Vl - A;2 sin^a; - Vr^A:^ 
 
46 TRANSCENDENTAL FUNCTIONS 
 
 xdx 
 
 /v cix 
 -;——. = - a; tan |a TT - a;) + 2 log cos i (i tt - x). 
 1 + sin X ^ 
 
 /cc doc 
 ' : = X ctn -^ (^ TT — x) + 2 log sin i (i tt — x). 
 J. olU X 
 
 331. iz = a; tan 4- cc + 2 log cos -J- a;. 
 
 J 1 + cos X 
 
 332. I :: = — x ctn ix + 2 log sin 4- x. 
 
 J 1 — cos a; 
 
 ««« r tan a; fZa; 1 .( ^h — a \ 
 
 333. , ==— =^=cos-M p— -cosa;)- 
 
 ^ Vet + ^ tan^ X -\/b — a \ V6 / 
 
 334. r-^^^^- = ^-yU-\^--tan-»( J-.tana- ) . 
 J a -\-b tan^ a; a — b\_ ^a \* /J 
 
 ««,. r tana^c^x 
 335. I 
 
 a + h tan x 
 
 = — — \bx — a log {a + b tan x) + a log sec a; V 
 
 336. I a; sin a;fZa; = sin x — x cos x. 
 
 337. I a;^ sin xdx = 2x sin a; — (a;^ — 2) cos x. 
 
 338. I a;^ sin a; c?a; = (3 a;^ — 6) sin x — (x^ — 6x) cos x. 
 
 339. J x"" sin xdx = — x™ cos x + m % x™~^cos xc?x. 
 
 340. I X cos xdx = cos x + x sin x. 
 
 341. I x^ cos X (fx = 2 03 cos x + (x^ — 2) sin x. 
 
 342. I x' cos X cZx = (3 x'' — 6) cos x + (x^ — 6 x) sin «. 
 
TRANSCENDENTAL FUNCTIONS. 47 
 
 343. I iC" COS xdx = x'" sin x — in i x'"~^sin xdx. 
 
 .... rsincc , 1 sin ic , 1 Tcos cc , 
 
 344. I dx = ■ —-—J H 7 I ——-7 dx. 
 
 J x"* m — 1 x'"-^ vi — lJx'"-^ 
 
 «.., rcosx , 1 cos a; 1 rsincc , 
 
 345. I dx = T ■ T T I 7«a;. 
 
 J X'" m - 1 a;'«-^ m - 1 J ic'"-* 
 
 346. J c?a; = a; - 77-77^ + ^^^ - ^^^7-, + 
 
 ' x^ 
 
 X 3-3! '5-5! 7-7! 9-9! 
 
 ,6 a.8 
 
 «>.« rcos X , , x^ , X* a;" , 
 
 3^^- J -^^^' = ^°^^-2:2!-^4T4!-6T6l + 8.8! 
 
 rxdx _ . x^ 7x^ 31a;^ 127 a-« 
 
 ^^^- J sinx"'^'^3.3!"^3-5-5!"^3-7-7!"^3-5-9!"^ 
 
 r^^ _ ^ g;^ 5.T^ 61^« 1385 a:^" 
 
 J cosa;~ 2 "*"4-2!"^6-4!'^8-6!"^ 10-8! ' 
 
 /x dx 
 . „ = — a; ctn a; + log sin x. 
 sin'' x 
 
 351. I — 5— = a; tan X + log cos a;. 
 
 J COS'^X 
 
 362. n^ I aj^'sin^xtfo; 
 
 = a;'"~^ sin"""^ x (m sin x — nx cos a;) 
 ' + w(?i — 1) rx^sin^-^XfZx — 7;i(w — 1) ra;"'-2sin"a;c?x. 
 
 353, 71^ J x"' CDS'* xc^a: 
 
 _ a;TO-i cos"~^ X (m cos x + nx sin x) 
 
 4- w(w - 1) fx'" cos'^-'^xcZx — m(?» - 1) Cx'"-'^ cos"xdx. 
 
48 TRANSCENDENTAL FUNCTIONS. 
 
 354. f^ 
 
 1 r a:!"*"^ (m sin x -{- (n — 2)x cos x) 
 
 ^ (?i - 1) (?i - 2) |_ sin«-ia; 
 
 365. f^ 
 
 »/ COS" a; 
 
 1 r x^~^(m cos g; — (?i — 2)a; sin a:) 
 
 ^ (?i - 1) (w - 2) L cos»-ix 
 
 ^ 'J cos"~^x ^ ^J COS" -^ a; J 
 
 ^^„ /'sin" re c?a; 
 356. J 
 
 X'" 
 
 X ((m — 2) sin x -\- nx cos a;) 
 
 ™m— 1 
 
 1 r sin"-^ 
 
 ~ (m - 1) (7n - 2) L 
 
 - n'j —;;;=r- + "(« - 1) J ^..-^ J 
 
 «^-. . 'cos" a:c?a; 
 357 
 
 1 P cos"~^ a; (na; cos a; — (m — 2) cos x) 
 
 ^ (m - 1) (??i - 2) L ^'""^ 
 
 „ /*cos"xfZx , . ., ^cos"~^xdx~\ 
 
 — ^ I 5 \- n(n — 1) I 7, — • 
 
 J a;'"-^^ ^ V a;'"-'' J 
 
 358. I x'' sin"* a: cos"a:c?x 
 
 = a:^"' sin™ a: cos"~^a;(» cosa; -j-(')/i + n)x sin a;) 
 
 (in + 7i)'' |_ \i \ 
 
 + (n — 1) (//I + ?i) I X'' sin^x cos"~^a:(^x 
 
TRANSCENDENTAL FUNCTIONS. 49 
 
 — m'p I a;''"^ sin^^^a; cos"~^a;c?ic 
 
 — p(^ — 1) I x^''"^ ^v^'x Q.o^'^xdx • 
 
 = •; — ; — -5 a:^""^sin"'~^a;cos"a:;(» sin a; — (w + w^a; cosa?) 
 (m + ny |_ v^ \ / / 
 
 + (m — 1) (m + 11) I aj^ sin"*"- a; cos" a;c?a; 
 -{-np i xP~^ sin"'~^x cos'^'^xdx 
 
 „-n C ■ ■ 7 sin Cm — n) a; sin(m + w)a; ^ 
 
 359. I sin mx sin nx ax = — ^r-^ i ^77 — ; — r~ * 
 
 J Z(m — n) Z{7n -\- 71) 
 
 __- /* . - COS (m — n)x cos(m + n)x g, 
 
 360. I sin mx cos nxdx — tt, r —rr) ; — . ' " 
 
 J 2(m — n) 2 (771 + 71) '^ 
 
 » 
 
 03 
 
 -_- /* , sin (m — w) a; , sin(m + w)a; 
 
 361. I cos waj cos wa; rfa; = — T-7 r 1 777 ; — f— • 
 
 ,/ 2 (m — 7i) 2(m + 71) •-' 
 
 362. I sin^ mxdx = tt — ("ma; - sin mx cos ma;). 
 »/ 27n^ ' 
 
 363. I cos"^ mxdx = pr — ("mx + sin mx cos mx). 
 ./ 2 ??i ^ '^ 
 
 1 
 
 364. I sin mx cos mxdx = — -f^ cos 2 mx. 
 J 4m 
 
 365. I sin nx sin"'x<Zx = — ; — — cos nx sin^'x 
 J m + 71 1_ 
 
 + m I cos(/4 — l)x -sin^'^xc^x 
 
50 TRANSCENDENTAL FUNCTIONS. 
 
 366. I sin nx Gos"'xdx = ; — — cos nx cos™ a; 
 
 J m + n [_ 
 
 + ml sm(7i — l)x-cos'''~^xdx \. 
 
 367. I cosnxsiu^xdx = ■ — sinna; sin'"a; 
 
 J m + n |_ 
 
 -«/sin(„-l)x.sin"-..<^.]. 
 
 368. I cos nx co^'^xdx = ; — sin nx cesser 
 
 J m + n\_ 
 
 + m I cos (w — 1) a; • cos"'~'icc?a; • 
 
 369 
 
 /cos nxdx _ /^cos {n — V)x dx /^ cos(w — 2)xdx 
 cos"'a; "" J cos"'~^a; J cos"'a7 
 
 rcosnxdx __ ^ r sin(?i — l)a;cZa; /* cos (?^ — 2)a-c?g; 
 ' J sink's; ~ J sin'"~ia; J sin'" a; 
 
 /'sin wa;fZa; _ ^ /^ cos(?i — l)a;c?a; r sin(??- — 2)a;6?a; 
 ' J sin"'a; J sin'"~^a; J sin"'a; 
 
 /'sin ??,a'fZa; _ ^ /' sin (?i — l)xdx /' sin (n — 2)a;c?a; 
 " J cos'" a; »/ cos'"" 'a; J cos'" a; 
 
 ^P + n-lfl^ 
 
 /'(cos ;:»a; + i s,\nj)x)dx _ . /';s'' 
 
 ■ J cos wa; »/ 1 + ;5;^" ' 
 
 where z = cos a; + t sin x. This yields two real integrals. 
 
 - . /'('cos wa; + i sin «a-)f7a; ^ rzP'^"~'^dz 
 
 374. I ^ ^— ^ = ~ 2 I — — -, 
 
 ^ sm na; J 1 — z^"' 
 
 where z = cos a; + i sin x. This yields two real integrals. 
 
TRANSCENDENTAL FUNCTIONS. 51 
 
 r(icosx — smx)dx_ r dy 
 375. J -J 2—;!' 
 
 where y = z^zzm This yields two real integrals. 
 
 V cos nx 
 
 X 
 
 «-,, r ■ • , • 7 , rcos(a — 6 + c) 
 
 376. I sm ax sm ox sin cxax= — x i ^ ; — ; 
 
 J \^ a — b + c 
 
 cos (b + c — a)x cos (a + b — c)x cos (a -\-b + c)x \ 
 6 + c — a a + 6 — c a + 6 + c J 
 
 'inn C z. J , fsin (a + ^> + c) 
 
 377. I cos ax cos te cos ca; aa; = i i ^^ — —-. — ; 
 
 J y_ a-\-b -\- c 
 
 sin (6 + c — ft) a? sin (rt — 6 + c) a? , sin (a + b — c)x \ 
 b ■\- c — a a — b -\- c a + b — c j 
 
 cos (a -^ b -h c)x 
 
 X 
 
 378. I sin aa? cos bx cos ca; c?a; = — :f i 
 
 cos (6 -f c — g) a! cos (a -\- b — c)x cos (a + c — i) x 
 
 } 
 
 b + c — a a + b — c a + c — b 
 
 ««« r -7 • 7 , r sin (a, -F ^» — c) a; 
 
 379. I cos ax sm bx sm ca;ax = f s ^^ — --. — 
 
 J I a -\- b — c 
 
 sin (a — b -h c)x sin (a + b -\- c)x sin (Z* -f- c — a.) a; 
 a — 6 + c a -^ b -\- c b + c — a 
 
 380. I sin~^a;c?a; = x sin~"^a; + Vl — x"^. 
 
 381. J COS'^XC/X = X QQS~^X — Vl — x^. 
 
 382. I tan-^xc/'a; = x tan~^x — \ log(l + x^). 
 
 383. j ctn- ^xdx = x ctrr ^ x -}- -^ log (1 + x^). 
 
x\ 
 
 62 TRANSCENDENTAL FUNCTIONS. 
 
 384. I s.eor'^xdx = x sec~^x — log(x + Vcc^ — 1). 
 
 385. I csc~^ a;c?x = x csc~'a; + log(a; + ■\f^— 1). 
 
 386. I versin-^ icci^a; = {x — 1) versin"' x + V2 a; — 
 
 387. C {?.m-^xfdx = a; (sin- ^ a;)^ - 2a; + 2 Vl - a;^ sin- 'a;. 
 
 388. I (cos-^ a;)^c?a: = x (cos"^ a;)^ — 2 x — 2 Vl — x^ cos~^ x. 
 
 389. fa; sin-^x^x - i[(2x2 - l)sin-ix + x Vl - x"]. 
 
 390. I X cos~^xc?x = i[(2x^ — l)cos~'x — xVl — x^]. 
 
 391. I X tan""'xc?x = ^[(x^ + l)tan~'x — x]. 
 
 392. I X ctn-'xc?x = ^[(x" + l)ctn-'x + x]. 
 
 393. I X sec~'xc?x = ^^[x^ sec""^x — Vx^ — 1]. 
 
 394. I X csc-^xc?x = ^[x^ csc~'x +a^'x'^ — 1]. 
 
 395. I x"sin-'xc?x = — — r ( x" + ^ sin-^x - \ , ^^ \ 
 J n-\-l\ -^ Vl - xV 
 
 396. I x"cos-'xcZx = — -— I x" + icos-'x 4- \ ^ , 
 
 J n + \\ J Vl -x^ 
 
i 
 
 ■^>-ck^ ^ ^---e'"' 
 
 ^^'■rc^ 
 
 TRANSCENDENTAL FUNCTIONS. 53 
 
 397. Jx"tsin-^xdx = ——(x'' + HEin-^x-Cj 
 
 398. Cx»ctn-'xdx = —^(x" + 'ctn-'x-^ f^!^l^Y 
 
 X / iC 
 
 400 
 
 tan~^a;c?a; , .,.-..„. tan~^a; 
 
 ^^ = logo; - 1 log(l + x^) - 
 
 X 
 
 401. Ce'''=dx = —- Cf{e"^)dx=J'^^^^^^, y^e'^. 
 
 a;e«^rfa; = — (ace - 1). 
 
 x^e'^'dx = I x'^-'^e"^dx. 
 
 a a J 
 
 P^ax 1 r e"^ , c^'^d^~\ 
 
 404. I — dx — : + a I ' 
 
 J x"" m — 1(_ x""-^ J a:"'-ij 
 
 J log a (log ay (log a)^ 
 
 n(n-l)(n-2)- • '2.1a^ 
 ~ (loga)" + i 
 
 a 
 
 bx 
 
 -«« Ta^'^a; 1 r «■" a^- 
 407. I — — = T --r - 7 
 
 log a 
 
 2)a;»-2 
 
 a^ • (log ay 
 
 + 
 
 (w-2)(ri-3)a;"-3 (w 
 
 (logq)"-^ r a='dx ~\ 
 
 - 2) (n- 3) "-2.1 J x J" 
 
 ,^„ Ca^dx , , , , (cc log a)^ , (a; log a) 
 
 3 
 
 + 
 

 J 
 
 54 TRANSCENDENTAL FUNCTIONS. 
 
 409. I :: = Ior i ^-^^ ' ■*- 
 
 J 1 + e^ '' 1 + e^ 
 
 /dx 1 
 411. \ —z;z-^-, — -: = ^=tan-M e^-^A/Tr 
 
 412. )— ^^== j=\\og(-\/a + be""'-Va) 
 
 / /— 2 -\/n 4- /)^'"^ 
 
 - log (Va +6 e"- + V^) L or == tan- ' ,Z_1 ^ 
 
 VI -y/ZTa V- a 
 
 J {\+xf 1 +x J a(n + 1) 
 
 A-iA C ^^ 7 e'^ia sin ?;.x — » cos px) 
 
 414. I e"" sm »a? (/a? = — ^^ -^ f ^-^• 
 
 Atn C r,^ 7 ^'^ (f' cos px + » sin »a^) 
 
 415. I e"" cos »x- dx = — ^^ ^, — ^ ^-^ • 
 
 416 
 
 e*" log a; c^a; = 2 I 
 
 a aJ X 
 
 goa: gij^2 xdx = J— — ^ ( sln X (o, B\Xi X — 2 cos cc) + - ] ' 
 
 /e'" / 2\ 
 
 goa; cos'^ajc^a; = — — — - ( cos a; (2 sin x + a cos a;) + - 1 • 
 
 419. I e'^sin"6xc?x = -5— — ;-^( (a sin bx 
 J a^ + tr¥ \ ^ 
 
 — nb cos bx) e'^ sin"~^ Jx + n (n — l)b- i e^ sin"~^ bxdx 
 
TRANSCENDENTAL FUNCTIONS. 
 
 55 . 
 
 ). I e'" cos" bxdx = -^—, — ^-^ ( (a cos bx 
 J a^ -\- n-h' \ ^ 
 
 + nh siu te) e"^ cos"~^ hx -\- n {n — X)h^ I e°^ cos"~26icc?a; )• 
 
 421. re^^tan^ajc^ic 
 
 n 
 
 
 /' 
 
 e'^tan"-ixc?a; — | e'^tan"-2a;c?x 
 
 422. re''^ctn«a;(Za; 
 e'^ctn"-!^ 
 
 423 
 
 / 
 
 n-1 
 
 e"^ dx 
 
 + 
 
 a 
 
 n 
 
 -J 
 
 -/' 
 
 e'^ctn''-^xdx— | e'" ctn"-^ a; c?a;. 
 
 a sin X -\-(n — 2) cos a? 
 
 pOX V i 
 
 sin" X {n — 1) {n — 2) sin''"^ x 
 
 + 
 
 a^ + (71 - 2y re^^dx 
 
 (n - 1) (7^ - 2) 
 
 -2) J sin"-^ 
 
 x 
 
 424. 
 
 / 
 
 e"^dx 
 
 ^ a cos X —(n — 2) sin a; 
 cos" a; "^ (n — 1) {n — 2) cos""~^aj 
 
 a^ + (71 - 2)^ r e'^'dx 
 
 + 
 
 -2^ f 
 
 -2) J ( 
 
 (» — 1) (ti — 2)»/ cos""~^a; 
 
 426. I e"^ sin"' a; cos"xdx 
 
 = 1 ; — TT"^ — ^ 1 ^""^ sin"^ X cos"~^ x (a cos x + (m + n) sin a;) 
 
 (w + ny + a^ K. ^ \ / / 
 
 — 7na I e'^sin"'~^a; cos"'~^a;(^a; 
 
 + (?z. — 1) (m + n) I e"^ sin"' a; cos"~'^a;c^a; V 
 
66 TRANSCENDENTAL FUNCTIONS. 
 
 = ~, ; — r^~. — :, \ e"^ sin"*"^ x cos" x (a sin x — (m -\- n) cosx') 
 
 (m + ny -\- a^ {. ^ ^ ' ' 
 
 ■\- na \ e'^sin'"~^x cos"~^icc?a; 
 
 + (m — 1) (to + ii) I e"^ sin"*"- x cos" xc^x \ 
 
 = •:^ 7^; -c\ re"^cos"""^a;sin"'~^a;('asina;cosic + ?isin^a; 
 
 (m + ny -\- a V 
 
 — mcos^a;)] + 7t(w — 1) | e"^ sin'"cccos"~^icc?a; 
 -\- mim — V) I e"^ sin'"~^ X cos^iccZo; [- 
 
 = -A re"'^sin"'~^£CCOs"~"^a;(asinxcosa:+ wsin^x 
 
 — m cos^x)] + w(w. — 1) I e"^ sin'"~^iccos"~^a;<Za; 
 
 + (m — n) (m + n — 1) j e"^ sin"'"^ x cos" xc?x V 
 
 = -^\ f(e''*sin"'~^a;cos"~^a;(asinxcosa; + nsin^x 
 
 {m + ny + a^ [^ ^ 
 
 — mcos^a;)] + m(m — 1) j e^^sin^^^xcos^^-xc^x 
 
 — (m — w) (w + w — 1) I e'"sin'"x cos"~^xo?x V . 
 
 426. I log xdx = X log X — X. 
 
 427. rx'"logx(Zx = x"' + ir^^^-- — ^^--^1- 
 
 428. j (log xydx = X (log x)" — 7i j (log x)"-^ c?x. 
 
 /x'""'' Vlos xV 7? /* 
 x™ (log x)" fZx = \ V T I a;'"(logx)"-'rfx. 
 ^ ^ m. 4-1 m + l»/ 
 
TRANSCENDENTAL FUNCTIONS. 57 
 
 (log xydx _ (loga;)""'"^ 
 
 430. f - ^. 
 
 J X /i + 1 
 
 ^3lJl|^-log(log.)^Iog.^(^V(|ifV 
 
 432 f ^^^ = ^ + _^f^ 
 
 J (log xy {n - 1) (log xy-^ n-lJ (log a;)"-' 
 
 r x"'dx a:"' + ' w +1 T a;"'c?a; 
 
 J (iog^~~(»i-l)(logx)»-i ?i-lJ (ioga:)"-i" 
 
 ■- — ^ = I dy, where ?/ = — (m + 1) log x. 
 
 log a; J V 
 
 435. r ^^ -\o.(\o<rx) and f C^^-D^--^ ^ -^ 
 
 J xlo^x- ^°^ ^^°^ ''^' J X (log X)" (log a;)»-i 
 
 436. flog («2 + cc2) (^a- = a; ■ log {a^ + x'')-2x + 2a- tan-' (^ ^ 
 
 437. j (a + Ja?)'" log a;6?a; 
 
 438. j a;"' log {a + to) o?a; 
 
 1 r /'a^'" "•" ' dx~\ 
 
 = ^^M^T r"" ^"^^" ^ ^"^ ~ ^ J "^Tto J * 
 
 /^log (g + ^3^) dx 
 
 439 
 
 X 
 
 , to 1 fbx\ , 1 /toV 
 = loga.logx + --2i(^-J +3i(^-; - 
 
68 
 
 TRANSCENDENTAL FUNCTIONS. 
 
 r \ogxdx 
 **"• J (a + bxy 
 
 441 
 
 _^ 1 r log a; r dx "I 
 
 6 (m — 1) |_ (a + te)"'-i J cc (a + te)"'-ij 
 
 /loa: xdx 1 , , , , ^ 1 /*log (a + ^ic) dx 
 
 — ^ = ^logx.log(« + fa)-J X 
 
 442. I {a + bx)\ogxdx=- — log a; ^-f ax — :^to^ 
 
 443. I ,^ 
 
 = - (log a; — 2)Va + bx +Va log(Va + 6x -fVa) 
 — Va log ( Va + 6a; — Va) , if a > 
 
 = - [(log a; - 2)Va + 6a; + 2V^ tan" ^ ^^^^^^ 1 .if a < 0. 
 
 444. I sin log xdx = ^x [sin log a; — cos log a^]. 
 
 445. I cos log a;c?a; = \x [sin log x + cos log a;]. 
 
 446. I sinh xdx = cosh x. 
 
 447. I cosh xdx = sinh a;. 
 
 448. I tanh a; c?a; = log cosh a;. 
 
 449. I ctnh xdx = log sinh x. 
 
TRANSCEIJTDENTAL FUNCTIONS. 59 
 
 450, I secli xdx = 2 tan~ ^ e'. 
 
 451. I csch a;c?a; = log tanh -• 
 
 /I . % — 1 /* 
 sinh"x(Zx = -sinh"~^a;-cosh a; I siuh"^^ xc?a; 
 
 sinh" + ^ a; cosh cc — r I sinh" + ^ a; ^x. 
 
 7i + 1 w + 
 
 /I . w — 1 /* 
 cosli"xc?a; = -sinha:- cosh""' a; H I cosh"~2a;^x 
 w n J 
 
 = sinh a; cosh" + ' a; H -^ | cosh" + ^ a; c?a;. 
 
 n + 1 n-^lJ 
 
 454. I x sinh xdx = x cosh a; — sinh x. 
 
 455. I a: cosh xdx = x sinh a? — cosh x. 
 
 456. j a;'' sinh xdx = (x' + 2) cosh a; — 2 a; sinh x. 
 
 457. I a;" sinh xdx = x" cosh a; — wa;""^ sinh a; 
 
 + n(n — 1) I a;"~^ sinh xdx. 
 
 458. I sinh^ a; c?a; = |^ (sinh x cosh x — x). 
 
 459. I sinh a; • cosh xdx = \ cosh (2 a). 
 
 460. I cosh^ a;(Za; = | (sinh x cosh a; + a;). 
 
 461. I tanh^a:c?x = a; — tanh x. 
 
60 TRANSCENDENTAL FUNCTIONS. 
 
 462. I ctnh^ xdx = x — ctnh x. 
 
 463. j sech^ xdx = tanh x. 
 
 464. j cscli^ic c?a; = — ctnh x. 
 
 465. I sinh~^ xdx = x sinh~' x — Vl + x^. 
 
 466. I cosh~^ a;c?a; = x cosh~^ x — Va- — 1. 
 
 467. j tanh- ^ cc (Za; = x tanh"^ a; + |- log (1 — a;^). 
 
 468. Cx sinh-i rrt/cc = ^[(2 x^ + 1) sinh"^ x - cc Vl + x^]. 
 
 469. I ic cosh-^ a;6?a; = :J[(2a;^ — l)cosh-^ x — xVx^ — 1]. 
 
 * ./ cosh a 
 
 + cosh a; 
 
 = csch a [log cosh ^ (a; + a) — log cosh ^ (a; — a)]. 
 = 2 csch a • tanh-^ (tanh ^ a- • tanh ^ a). 
 
 /dx 
 ; -. — = 2 CSC a ■ tan- ^ (tanh 4- a; • tan i^ a). 
 cos a + cosh X \ ^ ^ / 
 
 /dx 
 - — \ ; — = 2 CSC a • tanh" ^ (tanh ^ x ■ tan A a). 
 1 + COS a ■ cosh X \ i ^ / 
 
 473. j sinh x ■ cos x c?a; = ^ (cosh x ■ cos a; + sinh x ■ sin x). 
 
 474. I cosh X • cos xdx = ^ (sinh a; • cos x + cosh a; • sin x). 
 
 475. j sinh x- sin a;c?a; = ^ (cosh a; • sin x — sinh x • cos x). 
 
TRANSCENDEiSTTAL FUNCTIONS. 61 
 
 476. I cosh X • sin xdx = \ (sinh x • sin x — cosh x ■ cos a). 
 
 477. I sinh (ma;) sinh (wa;) c?a; 
 
 = — ^ 7, m sinh (?ia;) cosh (pix) — n cosh (jix) sinh (mo;) • 
 
 478. I cosh (mx) sinh (wa;) dx 
 
 = — ^ J m sinh (na;) sinh (wa;) — n cosh (wa;) cosh (vix) • 
 
 479. I cosh (mx) cosh (raa;) 6?x 
 
 = - 2 _ — ^ m sinh (ma;) cosh (wa;) — n sinh (/ia;) cosh (mx) ■ 
 
 / ■ dx _ r (Z(tana-) 
 
 a cos''^ X -\- c sin x • cos x -\- b sin^ a; J a + c tan a; + 6 tan"'^ ar 
 /(I + 7/1. cos a; + w sin x) dx _ T (m cos 8 + ?i sin 8) cos s • dz 
 a -{- b cos a; + (^ sin x J Z 
 
 ' I ■ dz r (7)1 sin 8 — n cos 8) sin s ■ dz 
 
 + . , . ^ 
 
 where b — q • cos 8, c = q- sin 8, s = a; — 8, Z = a. + y • cos 2;. 
 
 C . , , X • V , 7^ 7 [See 303 and 304.1 
 
 I sm (mx + a) • sm (na: + 0) aa; ^ -^ 
 
 sin [?«a; — nx + a — &] sin [?/ia; + '^^ + <* + ^] 
 
 ~ 2 (??i — 7i) 2 (?/i + «) 
 
 I cos (mx + a) • cos (iix + i) rfa; 
 
 sin [^mx -{- nx -\- a -\- b"^ sin [mx — nx -\- a — b"] 
 
 ^ 2 (wi + n) ^ 2 (??i - n) 
 
 I sin (mx -\- a) ■ cos (nx + h) dx 
 
 cos [ma- + wa^ + «■ + ^] cos [^mx — nx + a — b~\ 
 
 ~ 2 {m + w) 2 (?/i — w) 
 
62 MISCELLANEOUS DEFINITE INTEGRALS. 
 
 VI. MISCELLANEOUS DEFINITE INTEGRALS* 
 
 481. J x''-'^e-''dx= j log- dx = T(n). 
 
 T(z + l)=z-T(z), if z>0. 
 
 r(jj)-T(l-y)=^yiil>y>0. r(2)=r(l)=l. 
 
 r (w + 1) = n I, if n is an integer. T (z)= Il(z — 1). 
 
 r(i) = Vt^. Z(y) = i)^[log r(t/)]. Z(l) = - 0.577216. 
 
 >,oo C w^ X ,, r" a:'"-^^/:^- T(m)T(n) 
 
 482. a:'"-'(l -a;)"-irfa;= I -— — — — = ^\ , .-^ • 
 
 483. I sin"a;c?a;= ) cos"xdx 
 
 %/o «/o 
 
 484. 
 
 1-3-5 •• -(71-1) IT .^ . . , 
 
 = o A r> — . s -77' if w IS an even integer, 
 J • 4 • D • • ' in) Ii 
 
 = ■ ^ _ ^j if 71. is an odd integer, 
 
 = \ Vtt — ^ {-•) for any value of n greater 
 
 rTl + lJ than-l. 
 
 J'^'^sinmxc?ic 7r.„ ^^^.„ ^ 7r.„ ^. 
 = -) if m>0; 0, if m = 0: — — > it ??i<0. 
 a; .^ J 
 
 * For very complete lists of definite integrals, see Bierens de Haan, Tables d'inti- 
 grales (Ufinies, Amsterdam, 1858-64, and Nouv. Tables d'intigrales difinies, Leyden, 
 1867. 
 
MISCELLANEOUS DEFINITE INTEGRALS. 63 
 
 ,^^ /^* sin x» COS ma;c?a; ^ .- ^ ^ ^ ^ 
 
 485. I = 0, if w<- 1 or m>l; 
 
 »/0 X 
 
 — > ifw = — 1 or m = l: — > if — l<7n<l. 
 4 Z 
 
 .__ C^ &Ui^xdx TT 
 
 486. I ^ — = TT • 
 
 Jo x^ 2 
 
 487. J cos{x^)dx = J siii(a;2)(Zx = i\|-- 
 
 sin A;a; • sin mx dx = \ cos A:a; • cos mx dx = 0, 
 
 */o 
 
 if A; is different from m. 
 
 489. I sin^ mxdx= I cos^mccc^a; = jr* 
 
 .-- r'^ COS mxdx TT ^ ^^ 
 
 »/o 1 + a;^ 2 
 
 .-, f"^ eosxdx C^ smxdx ir 
 
 491.1 ^z^= I -- = ^-. 
 
 492. r"e-"'^^x = ^V^- = j^ra). 
 
 c/o 2 a 2(z ^^-^ 
 
 493. I a;"e-°^c?a;= ^ ^. ^ =^-7- 
 
 .«>. r" , » , 1-3-5- • •(2?i-l) Pr 
 
 Jo 2" + ia» ^a 
 
 e ^dx = ^ ^^ - a>0. 
 
 496. I e-"^ VxcZa; = 77- \/- • 
 «/o 2 n ^ 7i 
 
 497. f"^c^a; = V-- a>0, 
 Jo Vx ^ ^i 
 
64 MISCELLANEOUS DEFINITE INTEGRALS. 
 
 dx IT 
 
 498 
 
 
 
 •f 
 
 499 r*__^^^_^ 
 
 sinli (ma;) • sinh (nx) dx = \ cosh (ma;) • cosh (nx) dx 
 
 = 0, if m is different from n. 
 
 cosh^ (mx) dx = — \ sinh^ (mx) dx = 
 502. I sinh (mx) dx = 0. 
 cosh (mx) dx = 0. 
 sinh (mx) cosh (wx) c?a; = 0. 
 sinh (mx) cosh (mx) c?a; = 0. 
 
 506. I e~ "^ cos mx dx = -5— ; ? if a >■ 0. 
 
 a^ + m^ 
 
 
 
 a-* + m^ 
 
 J'' m 
 e-"^ sin mxdx = -r-; ;» if a > 0. 
 a-* + m"^ 
 
 6-'^'=^ cos hxdx = -^ a>0. 
 
 Za 
 
 ••X'l^ 
 
 509. I ^:^^^c;a; = -^- 
 x o 
 
 510. rM£^=_^. 
 
 511, r'J<^,&=-^^ 
 
 »/o 1 — a;'' 8 
 
MISCELLANEOUS DEFINITE INTEGRALS. 65 
 
 1 + x\ dx ir^ 
 
 512. f\og(l±^).^ = 
 
 »/o \1 — X/ X 
 
 513. r^i^l^ = -?log2. 
 Jo Vl-a;2 2 
 
 515. J (loga;)"cZa;=(- l)«-w!. 
 516.X'(lo.i)'.. = ^. 
 
 518. f , '^ = V?. 
 
 519. ra:"'log(-)(Za:= T'^''t,2i >^f^ + 1>0, ^ + 1>0. 
 
 520. flog ('?^V = T- 
 Jo ^ \e^ - ly 4 
 
 jr IT 
 
 log sin xdx = \ log cos a;c?aj = — — • log 2. 
 .0 c/o 2 
 
 X ■ log sin xdx = — — log 2. 
 523. I log (a±b cos x)dx = tt log ( ;r ] • « ^ 6. 
 
66 ELLIPTIC INTEGRALS. 
 
 VII. ELLIPTIC INTEGRALS. 
 
 d6 r^ dz 
 
 where k^ <d, x = sin <fi. 
 E {4>, k)=f Vl - k^ sin^ e ■ dO. 
 
 y 
 
 (1 + ri sin^ ^) Vl - k' sin^ ^ 
 
 <^ = am u, sin <^ = cc = sn m, cos <f> = Vl — aj^ = en m, tan (f> = tnu, 
 A<t> = Vl - A;2 sin^ <^ b Vl - kV = dn «, A;'^ = 1 - A^l 
 
 t< = am~^(<^, ^)=sn~^(a;, A;)=cn~^(Vl — x% k) 
 = dn-i(Vl-;k2^2, A;). 
 
 JS:=i^(i7r, ^), K'=F(i7r, k'), E=E{^'ir, k), E'-E^Tr, k'). 
 
 T* 7 2 A;* sin 2 (0 
 
 It ko = q — — 7 and tan <^ = 
 
 1 + k k + cos 2 <i) 
 
 524. r '' 
 
 »/0 
 
 Vl - k^sin^e 
 
 = I [i + (i)'.' + [^) k' + (ifl)"^ +.,.], it ,. 
 
 <1. 
 
 = K. 
 
 525. Vl- k'sin'^ede 
 
 ). J Vl 
 
 =i[-(«--(i^yf-G^yf--]""'<^- 
 
ELLIPTIC INTEGRALS. 67 
 
 ,474 X'O'O A -J c. I I 
 
 = J'\<i>, k), 
 
 3 5 5*3 
 
 where VI4 = i sin^ «^ -f- — , ^ = ^ sin* <^ + — sin^ <^ + ^^^, 
 
 A = isin«<^ + g^sin*<^ + g^sin^<A + 3^g^ 
 
 . r* Vl - k^ sin' 6- dd = - <i>- E -\- Bva <!> cos, A ~k' 
 
 1 
 
 = E{4>, k). 
 
 J , = sn-i(aj, k) 
 
 V(l - a;2) (1 - k'^x') 
 
 528. . , 
 
 V(l - x") (1 - k'^x') 
 
 = F{sh\-^x,k). Q<x<l. 
 
 529. r , '^"^ =cn-^(g, A;) 
 »^^ V(l - x^) (A;'2 + Ji-x^) 
 
 = F{cos-^x, k) = sn-i ( Vl - x% k). 0<x<l. 
 
 530. C , "^^ =dn-i(a;, A;) 
 
 = #(A-ia;, k) = sn-^ Q Vl - x% k\ < a; < 1. 
 
 531. r , ^"^ =tn-\x,k) 
 -'o V(l + X') (1 + A;'2a;2) 
 
 = i^(tan-'a;, A;) =sn-Y--=^=' A; Y < cc < 1. 
 
 \ Vl 4- cc^ y 
 
 * The next forty-two integrals are copied in order from a class-room list of Prof. 
 W. E. Byerly. 
 
68 ELLIPTIC INTEGRALS. 
 
 532. r . ^^ =2sn-U^,k) 
 ^0 -Vx (1 -x)(l- Jc'x) ' 
 
 = 2F(sm-Wx,k). 0<a;<l. 
 
 533. r—==J^== = 2 cn-i ( V^, k) 
 ^^ Vx (1 - x) (k'^ + k^'x) ^ ^ 
 
 = 2 i^(cos-i V^, k) = 2 sn-i(Vl-cc, A;). < a; < 1. 
 
 534. r ^ ^^ = 2 dn-i ( V;, A;) 
 
 •^^ Vx (1 - a;) (x - k'^) ^ [ 
 
 = 2i^(A-i V^, k) = 2 su-i Q Vl -x,k\- 0<a;<l. 
 
 535. r , "^^ =2tn-^(V^, /^) 
 ^^'o V(l + a;) (1 + A;'2a;) ^ ^ 
 
 = 2i?'(tan-iV^, A;) = 2sn-Y^-^,>tY 0<a:<l. 
 
 536. r , ^^ ^lsn-Yf,-^Y a>6>a;>0. 
 
 537. r , ^^ ^lsn-Y^,^Y ^>a>^. 
 
 538. . 
 
 ^x V(a2 4- x") (b^ - x") 
 
 cn-if?, ^=i=Y b>x>0. 
 
 V^M^' V^ V 
 
 539. J^ ^^ 
 
 
J -a 
 
 ELLIPTIC INTEGRALS. 69 
 
 dx 
 
 V(a* - a;2) {x' - b^) 
 
 1 ,/ \a^ - x^ la'' - b^\ 
 
 541. r ^^ 
 
 '" V(x2 + ay(^M^') 
 
 542. X' ''^ 
 
 V(a; - a) (x - y8) (a; - y) . 
 
 y 
 
 
 Va-y K^^-y ^a-y 
 
 V(a; — a) (cc — /3) (a- — y) 
 2 
 
 Va^ 
 
 y 
 
 (V!^;- V!5^) 
 
 
 '^ V(a — x)(x — /3) (a; — y) 
 
 545, 
 
 
 Va — y \^«-^ 
 
 c?a; 
 
 V(a — a;) (a; — )S) (x — y) 
 
 2 ^..-i/'./^Lziy i^^^. J^-/3^ 
 
 Va-y V^«-/5^-y ^«-y. 
 
 c?a; 
 
 V(a^^^) (/? — a;) (a; — y) 
 2 
 
 Vo-" 
 
 a > a; > 6. 
 
 1^ ,/a; /a^ - b'\ 
 
 a > /3 > y. 
 
 ^sn-Ur^, V^^^V a.>a. 
 
 sn-M \ ^' \r^ ^ • x>a. 
 
 ' --('^/^»■ Vfi|)- <'>->^- 
 
 sn-i(-V ^ ^' -V ^ )• a>a->^. 
 
 Va — y 
 
 546. f^ 
 
 \ iS — y a — a; ^a — yj 
 
70 ELLIPTIC INTEGRALS. 
 
 '^ dx 
 
 5«X 
 
 y V(a — x){fi — x) (x — y) 
 2 
 
 sn" 
 
 548 
 
 
 Va — y 
 
 ■(^§i^. V!5^)- ^>^>y 
 
 ^ V(a — cc) (/3 — cc) (y — a;) 
 
 Va 
 549. f 
 
 7 
 
 (V^.- >/^) 
 
 sn-^UL:^, X^^-^ • v>a;. 
 
 V(a-x)(^-a^)(y-a;) 
 2 , / /a — y 
 
 Va — V \ ^a — a; 
 
 
 sn-M \ ^' \ ^ • y>x 
 
 550. 
 
 X' 
 
 a > y8 > y > 8. 
 dx 
 
 V(x - a) (cc - /3) (a; - y) {x - 8) 
 2 
 
 , / 113-8 X -a l/3-y a-8\ 
 
 V(a-y)08-8) 
 
 a;>- a, 
 
 531. f"- "^^ 
 
 V(a. — a;) (a; — /3) (a:; — y) (x — 8) 
 
 \^a — j8x — 8 ^a — 
 
 ^ y-8^ 
 
 552 
 
 V(a - y) (/3 - 8) \^a--^x-8 ^a-y/3-8j 
 
 a>x> fS. 
 
 ■X 
 
 ^ V(a - a;) (a; - ^) (a; - y) (x - 8) 
 
 \^a— ^x— y ^ a 
 
 2 _g^-ir J«zi^y ^-^. J«^i^ 1^ 
 
 V(a - y) ()8 - 8) \^a-^a;-y >'a-y^-8y 
 
 a > a; > /3. 
 
553 
 
 X 
 
 ELLIPTIC INTEGRALS. 71 
 
 ^ dx 
 
 ^ -si {a - x) (/3 - X) {x -i){x- 8) 
 
 V^/3-y a-x' ^a-y ^-l) 
 
 2 
 
 sn" 
 
 V(a - y) (/? - 8) 
 
 ^ > CC > y. 
 
 554. J''^ "^^ 
 
 ^ V(a - CI-) (^ - a;) (cc - y) {x - 8) 
 
 2 
 
 sn" 
 
 V^)8-y a;- 8 ^/a 
 
 ;8-y o^^ 
 
 ■V(a -y)()8-8) V^^-y^-S ^a-yj8-8^ 
 
 i8 > a; > y. 
 
 555. f^ "^^ 
 
 V(a -x){(3- x) (y - a;) (x - 8) 
 
 .2 s,,-irj^^ y^^. J^^:il y-8 
 
 Va 
 
 V(a -y)(/3-8) V^y-S^-a; >'a-y^-8y 
 
 y > a > 8. 
 
 556. J^ ^^-^ 
 
 's V(a - a;) ((8 - x) (y - x) (a; - 8) 
 
 \ *y — 6 a —X ^a 
 
 1 .^-if.l^nj^^^^^. J^l^lI y-g 
 
 V(a-y)(^-8) \^y-8a-x ^a-y(i-8j 
 
 y>x>8. 
 
 X^ (Ir 
 
 V(a - x){fi- x) (y - a;) (8 - a;) 
 2 
 
 sn" 
 
 i/^J^-y.g-^ /)8-y a-8\ 
 V^a-8 y-a;' ^/a - y )8 - 8/ 
 
 V(a-y)(/3-8) 
 
 8>a;. 
 
 558. i sna;c?x = - cosh~M -yp ]• 
 
 559. I en a; c?a; = - cos~^ (dn a;). 
 
72 ELLIPTIC INTEGRALS. 
 
 560. I dn xdx = sin~^ (sn x) = am x. 
 
 561. r-^^^iogf" 'Y^ 1- 
 
 J s,nx [_cn X 4- dn a; J 
 
 ^nn r (^^ It F^' SH 0? + dll X~\ 
 
 562. I = - log • 
 
 J cnx k' \_ en ic J 
 
 _„„ /* c?aj _1 J FA;' sn X — en a?"] 
 
 ' J dn X k' \_k' sn ic + en xj 
 
 sn^xdx = Ti[a^ — -E^(ania;, A;)], 
 
 565. j en^ajc^cc = — [^(ama;, A;) — k'^x"]. 
 
 566. I dn^a;c?a; = ^(am x, k). 
 
 567. (m + 1) fsn'^ajc^x = (m + 2) (1 + A;^) fsn'^+^a^cfaj' 
 
 — (m + S)k^ I sn^ + ^a^c^a; + sn'" + 'a; en a; dna;, 
 
 568. {m + l)k'^Ccn"'xdx = (m + 2) (1 - 2 k^)Ccn"'+^xdx 
 
 + (m + 3)A;M cn'"+ *a;c?a; — cn'"+^a; snxdnx, 
 
 569. (m + 1)^'2 rdn"»a;c?a; = (m + 2) (2 - k^)fdn'" + ''xdx 
 
 — (m + 3) j dn"*+*a;c?x + A;^dn"*+^a;snaena;, 
 
 Since sin2 ^ = _ _ _ (i _ A;2 • sm2 ^), 
 
 J ^2 sin2 6id& \ r^ dd 1 /^2 ^ 
 
 VI -A;3sin2(? ^Vo Vl _ A;2sin2(? *Vo 
 
TRIGONOMETKIC FUNCTIONS. 73 
 
 Vm. AUXILIARY FORMULAS. 
 
 A. — Trigonometric Functions. 
 
 570. tan a • ctn a = sin a • esc a = cos a ■ sec a = 1. 
 tan a = sin a -j- cos a, sec^ a = 1 + tan^ a, 
 csc^a = 1+ ctn^a, sin^ a + cos^ a = 1. 
 
 571. sin a = V 1 — cos^ a = 2 sin ^ a ■ cos ^ a = cos a • tan a 
 
 fe=Vi 
 
 1 tana /I — cos 2a 2tan4-a 
 
 Vl + ctn^a Vl+tan^a ^ 2 l + tan^^a 
 
 =v 
 
 gpp- ^ "1 
 
 = ctn ^a • (1 — cos a) = tan -^ a • (1 + cos a). 
 
 sec^a 
 
 572. cos a = Vl — sin^ a = = = = 
 
 Vl + tan^ a Vl + ctn^ a 
 
 -4 
 
 1 + cos 2 a 1 - tan^ ^a „ , . „ , 
 
 ~?^ = -. , ^ — rf — = cos^ia — sin^ia 
 
 2 1 + tan^ ^ a ^ ^ 
 
 = 1—2 sin^ |- a = 2 cos^ ^ a — 1 = sin a • ctn a 
 sin 2 
 
 a _ /csc^ ^ ~ 1 _ ^^^ i ^ — ^^'^ "2" ^ 
 a * csc^ a ctn 4- a + tan 4- a 
 
 -„n ^ sin a Vl — cos^ a sin 2 a 
 
 o7o. tan a = 
 
 Vl — sin^a cos a 1 + cos 2 a 
 
 1 — cos 2 a _ /l — cos 2 a _ 2 tan ^ a 
 sin 2 a ' 1 + cos 2 a 1 — tan^ ^ a 
 
 sec a _ 2 _ 2 ctn |^ a 
 
 esc a ctn \ a — tan ^ a ctn^ ^ a — 1 
 
74 
 
 574. 
 
 TEIGONOMETEIC FUNCTIONS. 
 
 1 — 
 
 — a. 
 
 90° ± a. 
 
 180° ± a. 
 
 270° ± a. 
 
 360° ± a. 
 
 sin 
 
 — sin a 
 
 + cos a 
 
 T sin a 
 
 — cos a 
 
 ± sin a 
 
 cos 
 
 + COS a 
 
 T sin a 
 
 — cos a 
 
 ± sin a 
 
 + COS a 
 
 tan 
 
 — tana 
 
 T etna 
 
 ± tana 
 
 T etna 
 
 ± tan a 
 
 ctn 
 
 — ctn a 
 
 T tana 
 
 ± ctn a 
 
 T tana 
 
 ± etna 
 
 sec 
 
 + sec a 
 
 T CSC a 
 
 — sec a 
 
 ± CSC a 
 
 + sec a 
 
 CSC 
 
 — CSC a 
 
 + sec a 
 
 T CSC a 
 
 — sec a 
 
 ± CSC a 
 
 575. 
 
 
 0°. 
 
 30°. 
 
 45°. 
 
 60°. 
 
 90°. 
 
 120°. 
 
 135°. 
 
 150°. 
 
 180°. 
 
 sin 
 
 
 
 i 
 
 i^ 
 
 iVs 
 
 1 
 
 iV3 
 
 iV2 
 
 i 
 
 
 
 COS 
 
 1 
 
 iVi 
 
 iV2 
 
 i 
 
 
 
 -i 
 
 -iV2 
 
 -IV3 
 
 -1 
 
 tan 
 
 
 
 1 
 
 V3 
 
 1 
 
 V3 
 
 00 
 
 -V3 
 
 — 1 
 
 1 
 V3 
 
 
 
 ctn 
 
 CO 
 
 V3 
 
 1 
 
 1 
 
 V3 
 
 
 
 1 
 
 V3 
 
 —1 
 
 -V3 
 
 00 
 
 sec 
 
 1 
 
 2 
 V3 
 
 V2 
 
 2 
 
 CO 
 
 -2 
 
 -V2 
 
 2 
 V3 
 
 -1 
 
 esc 
 
 CO 
 
 2 
 
 ^y^ 
 
 2 
 V3 
 
 1 
 
 2 
 V3 
 
 ^ 
 
 2 
 
 00 
 
 576. sin ^ a = V^(l — cos a). 
 
 577. cos ^ a = V^(l + cos a). 
 
 578. tan ^ a = ^— 
 
 cos a 
 
 cos a 
 
 sm a 
 
 + cos a sin a 1 + cos a 
 
 579. sin 2a = 2 sin a cos a. 
 
 580. sin 3 a = 3 sin a — 4 sin^ a. 
 
 581. sin 4 a = 8 cos^ a • sin a — 4 cos a sin a. 
 
TRIGONOMETRIC FUNCTIONS. 75 
 
 582. sin 5 a = 5 sin a — 20 sin^ a + 16 sin* a. 
 
 583. sin 6 a = 32 cos* a sin a — 32 cos^ a sin a + 6 cos a sin a. 
 
 584. cos 2a = cos^ a — sin^ a = 1 — 2 sin^ a = 2 cos^ a — 1. 
 
 585. cos 3 a = 4 cos^ a — 3 cos a. 
 
 586. cos 4 a = 8 cos* a — 8 cos^ a + 1. 
 
 587. cos 5 a = 16 cos* a — 20 cos^ a + 5 cos a. 
 
 588. cos 6 a = 32 cos^ a — 48 cos* a + 18 cos^ a — 1. 
 2 tan a 
 
 589. tan2a = 
 
 590. ctn2a = 
 
 1 — tan^ a 
 ctn2 a - 1 
 
 2 ctn a 
 
 591. sin (a±ft) = sin a • cos /? ± cos a • sin )8. 
 
 592. cos (a± fi) = cos a • cos yS =f sin a • sin ft. 
 
 ..«« , ^N tan a ± tan 5 
 
 593. ta.n(a±ft) = - — ^• 
 
 ^ '^^ 1 rp tan a ■ tan /3 
 
 ..«.. , ^x ctn a • ctn )S rp 1 
 
 594. ctn(a±/5) = — ^ ^ Z, ' 
 
 ^ ^^ ctn a ± ctn /3 
 
 595. sin a zt sin )8 = 2 sin ^ (a =b /3) • cos i(a + iS). 
 
 596. cos a + cos /8 = 2 cos ^(a + /8) • cos |(a - /3). 
 
 597. cos a - cos /8 = - 2 sin |(a + /3) • sin \{a- ft). 
 
 sin ("a d= S) 
 
 598. tana±tan/3 = ^ ^• 
 
 cos a • cos ft 
 
 -~^ « sin (a ± ft) 
 
 599. ctn a ± ctn ;3 = ± ^-^^ — r^- 
 
 '^ sin a- sin /3 
 
76 TRIGONOMETRIC FUNCTIONS. 
 
 _„_ sin a ± sin yS , , , _, 
 
 600. ■ ^ = tan i (a i S). 
 
 cos a + cos p ^ ^ 
 
 sin a dz sin /3 
 
 601. ^ = — ctn i (a + S). 
 
 cos a — cos /8 i \ » / 
 
 „-_ sin g + sin ^ _ tan -|- (a + j8) ^ 
 sin a — sin /3 tan ^ (a — j3) 
 
 603. sin2 a - sin^ ^ = sin (a -\- (3) ■ sin (a - (3). 
 
 604. cos' a - cos' ^ = - sin (a + /3) • sin (a - /3)c 
 
 605. cos' a — sin' /3 = cos (« + /3) • cos (a — /3). 
 
 606. sin xi = ^ i(e^ — e~^) — i sinh x. 
 
 607. cos xi = ^{e^ + e~^) = cosh x. 
 
 608. tan xi = -^^ — ; — i tanh x. 
 
 6^ + e~^ 
 
 609. e^+ 2'' = e^ cos ?/ + ie^ sin ?/. 
 
 610. a^ + 2'' = a=^ cos (2/ • log a) + ta'^ sin {y • log a). 
 
 611. (cos zti- sin ^)" = cos nd ±i- sin n^. 
 
 612. sin a; = — i i(e" - e"^). ^ 
 
 613. cos a; = I- (e^" + e""). 
 
 614. tan x = — i -z—. • 
 
 e-^ + 1 
 
 615. sin (x ± yi) = sin x cos yi ± cos x sin yi 
 
 = sin X cosh y ± * cos x sinh y. 
 
 616. cos {x ± yi) = cos X cos 3/1 q= sin x sin yt 
 
 = cos X cosh y 4= » sin x sinh y. 
 
TRIGONOMETRY. 77 
 
 617. 
 
 In any plane triangle, 
 a b c 
 
 sin A sin B sin C 
 
 618. a'^ = h''^-c''-2bc(toQA. 
 
 „-_ a-\-b _ sin ^ + sin B _ tan ^ (A -\- B) _ ctn ^ C 
 ' a — b sin .4 — sin B tan ^ (.4 — i?) tan ^ (.4 — 5) 
 
 620. sin^^=^^^ — ^^ — ^, where 2s = a + 6 + c. 
 
 621. cosM=^pZ«). 
 
 622. tani^^>~/^^^;^> 
 ^ >' s (s — a) 
 
 623. Area = ^ ic sin ^ = Vs (s — a) (s — 6) (s — c). 
 
 In any spherical triangle, 
 
 -„. sin A sin B sin C 
 
 624. -^ = —. — - = — 
 
 sm a sin b sm c 
 
 625. cos a = cos 6 cos c + sin 5 sin c cos A. 
 
 626. — cos ^ = cos B cos C — sin B sin C cos a. 
 
 627. sin a ctn & = sin C ctn B + cos a cos C. 
 
 «r.o , i /sin s • sin (s — a) 
 
 628. cosi-^=-V • r, ■ -• 
 
 ^ ^ sin b ■ sm c 
 
 ««« . , A /sin (s — 6) • sin (s — c) 
 
 629. sinAJ = -V — '^ — ^I — ^^ 
 
 ^ ^ sm b ■ sm c 
 
 ««o. , . /sin (s — ^) • sin (s — c) 
 
 630. ta.ulA = \ ^ i , . ^ ' 
 
 ' ^ sm s • sm (s — a) 
 
78 TrwIGONOMETRY. 
 
 oQi 1 ^ j cos (S -B)- COS (S-C) 
 
 631. GOS^a=\ ^ : j- r-^-; ^- 
 
 » sin f» . sm f ; 
 
 sin B • sin C 
 
 „r,n ■ 1 — COS S-cos(S — A) 
 
 632. smia = V ■■ — „ • ^ ^• 
 
 ■^ \ sin « sm n 
 
 CQQ 4- 1 * / — COSTS' -cos (.S — ^) 
 
 633. tania = V eos(^-^).cos(^'-C) - 
 
 2s = a-{-b-\-c. 2S=A + B + C: 
 
 634. cos \{A-\-B) = ^^—^ '- sm \ C. ■ 
 
 coK 1/^ x)\ sin-|-(a + &) . 
 
 635. cos ^ (J — ^) = T-^^-r ^ sin ^ C. 
 
 sin -^^ c 
 
 636. siniM + 5) = 2^^^^^P^cosia 
 
 ^ -^ cos ^ c 
 
 637. sin \{A — B) = V^ cos \ C. 
 
 638. tani(^ + B) = 55ll|^etaia 
 
 639. tan«^-^) = ?|±i|^ctnia 
 
 640. tan |(a + ^') = ^"^ t ^^ 7 f x tan i c. 
 
 ^ ^ ^ cos |(^ + 5) ^ 
 
 641. tan ^(^ - b) = ^!" t /^ 7 ^x tan ^c. 
 
 g^2 cos ^(a + b) ^ ctn|C _ 
 cos ^(a — b) tan -^ (^ f ^) 
 
ANTITRIGONOMETRIC FUNCTIONS. 79 
 
 In interpreting equations which involve logarithmic and 
 anti-trigonometric functions, it is necessary to remember that 
 these functions are multiple valued. To save space the 
 formulas on this page and the next are printed in con- 
 tracted form. 
 
 643. sin-^a; = cos~^ Vl — x^ = tan~^ 
 
 X 
 
 — sec~^ 
 
 = CSC 
 
 X 
 
 1 - = 2 sin-i [^ - i Vl - a;2]i 
 
 = i sin-i (2 X Vr=T2) ^ 2 tan-» \^ — ^ | 
 
 = -^tan-M ^--^^-J=i7r-cos ^a; 
 
 = 4" TT — sin~^ Vl — x^ = — sin~ ^ (— a;) 
 
 = ctn-i^^^^!— ^^ = (2w-f^)7r-nog(a;+Va;2-l) 
 
 = i TT + ^ sin-\2 a;2 - 1) = ^ cos-'(l - 2 x^). 
 
 VT^-^ 1 
 
 644. cos~^x = sin'^ Vl — x^ = tan~^ = sec""^ - 
 
 X X 
 
 = -^TT — sin~^a; = 2 cos" 
 = |-cos-i(2a;2-l) 
 
 ■v^ 
 
 = csc~^ — ■ = TT — COS" V— aj) 
 
 Vl - a;2 ^ ^ 
 
 = ctn-i ~ 
 
 Vl 
 
 x* 
 
 i log (a; -f Va;^ — 1) = tt — t log ( V^^ — 1 — a;). 
 
80 ANTITRIGONOMETRIC FUNCTIONS. 
 
 645. tan~^cc = siri"^ — , = cos~^ , = h sin-^ -— — ; 
 
 X 
 
 = -J-TT — tan~^ - 
 x- 
 
 L 2 Vl + x2 J L 2 ViT^ J 
 
 = ^ Un- ,-1^, = 2 tan- [^ll+l^'] 
 
 1 — X^ |_ X J 
 
 = — tan~^ c + tan~^ :; = — tan~^ (— x) 
 
 [_1 — ex J ^ ' 
 
 = i * log TT^ — ; = i * log -. 
 
 646. sin~^ a; ± sin~^ y — sin"^ [cc Vl — if ±y Vl — a;*]. 
 
 647. cos~^ X zb cos~^ y = cos~^ [xy if V(l — x^) (1 — ?/^)]. 
 
 648. tan-i a; ± tan-^ v = tan-^ ^^J- i . 
 
 648. sin~^ a; ± cos~^ y = sin~ ^ [a:/y zt V(l — x^) (1 — ?/*)] 
 
 = COS" ^ [?/ Vl — x^ If a: Vl — y^]. 
 
 650. tan- ^ x ± ctn-^ y - tau- ^ f^^^l = ctii- ' [^^1 
 
 651. log (x + yi) = i log (x^ + y^) + i tan"" ' (jj /x). 
 
HYPERBOLIC B'UNCTIONS. 81 
 
 B. — Hyperbolic Functions. 
 
 652. sinh a; = ^ (e^ — e^^) = — sinh (— x) = — i sin (ix) 
 
 = (csch x)~^ — 2 tanh ^x -7-(l — tanh^ ^^)- 
 
 653. cosh X = ^ (e^ + e~^) = cosh (— a;) = cos (ix) — (sech ic)~^ 
 
 = (1 + tanh^ ^x)^(l- tanh^ ^ x). 
 
 654. tanh a; = (e^ - e"^) -^ (e^ + e"^) = - tanh (- x) 
 
 = — i tan (tx) = (ctuh x)~^ — sinh a; -^ cosh a;. 
 
 655. cosh xi = cos a;. t*^ z^ X ^ ^»*^V 
 
 656. sinh a;* = ^ sin a;. r57 -^^iX ^ eA-A/^^X 
 
 657. cosh^a; — sinh^a; = 1. 
 
 658. 1 — tanh^x = sech^a?. 
 
 659. 1 — ctnh^a; = — csch'^a;. 
 
 660. sinh (x ±y) = sinh x ■ cosh y ± cosh a; • sinh y. 
 
 661. cosh (x ±y) = cosh a; • cosh ?/ ± sinh a; • sinh y. 
 
 662. tanh (a; zt ?/) = (tanh x ± tanh ?/) ^ (1 rb tanh x • tanh ?/). 
 
 663. sinh (2 a;) = 2 sinh a; cosh x. 
 
 664. cosh (2 a;) = cosh2a;-|-sinh^a; = 2 cosh^a: — l = l+2sinh^a;. 
 
 665. tanh (2 a;) = 2 tanh a; ^ (1 + tanh^a;); 
 
 666. sinh (i a;) = V^ (cosh a; - 1). 
 
 667. cosh (J- x) = Vi (cosh a; + 1). 
 
 668. tanh (i a;) = (cosh x — 1) -i- sinh x = sinh a; -f- (cosh a; + 1). 
 
 669. sinh x -\- sinh y = 2 sinh -^ (a; + ?/) • cosh ^ (x — y). 
 
 670. sinh x — sinh y = 2 cosh ^ (x + 2/) • sinh ^(x — y). 
 
82 HYPERBOLIC FUNCTIONS. 
 
 671. cosh X + cosh y = 2 cosh ^ {x + y)- cosh ^(x — y). 
 
 672. cosh X — cosh y = 2, sinh ^ {x + y) ■ sinh ^{x — y) 
 
 673. d sinh a; = cosh x ■ dx. 
 
 674. c? cosh X = sinh a; • dx. 
 
 675. c^ tanh x = sech^ a; • dx. 
 
 676. 0? ctnh a; = — csch'^ x • c?a;. 
 
 677. d sech a; = — sech a; • tanh x • c?a;. 
 
 678. d csch a; = — csch x ■ ctnh a; • dx. 
 
 dx 
 
 679. sinh-'a; = log (a; +Va;2 + 1) = J" 
 
 Vx^ + l 
 
 = cosh~^ Va;^ + 1. 
 
 680. cosh- 1 X = log (a; + Vx^ - 1) = J 
 
 Va;2-1 
 
 = sinh-^ Vx^ — 1. 
 
 /rfa 
 ^-3 
 
 x^ 
 
 /rfx 
 dx 
 
 683. sech- ^x = log (^^ + yj^, _ 1^ = - J 
 
 684. csch'^x = log f- + yj^ + 1 ) = "X 
 
 X Vl — x^ 
 dx 
 
 685. c? sinh-^x = 
 
 686. c? cosh~^x = 
 
 xVx^ + l 
 <^x 
 
 Vl+X^ 
 
 dx 
 
 VV 
 
687. dta,nh-^x = 
 
 HYPERBOLIC FUNCTIONS. 83 
 
 dx 
 
 688. dGtnh-^x = - 
 
 689. dseGh-^x = - 
 
 690. dcsGh-^x = - 
 
 1-x^ 
 dx 
 
 x^-1 
 dx 
 
 ic Vl — x^ 
 dx 
 
 X 
 
 V^+i 
 
 If m is an integer, 
 
 691. sinh (mTri) = 0. 
 
 692. cosh {miri) = cos mTT = (— 1)*". 
 
 693. tanh (mTri) = 0, 
 
 694. sinh (x + mTri) = (—!)'» sinh x. 
 
 695. cosh (x + mTri) = (— 1)™ cosh (x). 
 
 696. sinh (2 m + 1) ^ Tri = ^ sin (2 m + 1) ^ tt = ± i 
 
 697. cosh (2 m + 1) i T^^ = 0. 
 
 -rr ±X ] = i cosh CC. 
 
 
 799. cosh (— ±a;j=±:i sinh x. 
 
 700. sinh w = tan gd u. 
 
 701. cosh u = sec gd m. 
 
 702. tanh u = sin gd u. 
 
 703. tanh ^ m = tan i gd w. 
 
 704. u = log tan (i tt + ^ gd u). Tsec a; r/rr = r/d' ' 
 
 a:. 
 
84 ELLIPTIC FUNCTIONS. 
 
 Elliptic Functions. 
 
 dz r* dO 
 
 
 V(l - s;2) (1 - A"2^=^ Jo Vl-A-2siii2^ 
 where A- <C 1, and x = sin <^, <^ is called the amplitude of u and 
 is written am (u, mod A;), or, more simply, am w; x = sin <fi = smc, 
 
 Vl — x^ = cos <^ = en w, Vl — A;^a;^ = A<^ = An r< = dn %, 
 
 Hence, am(0)=0, sn(0)=0, cn(0) = l, dn(0)=l, 
 am (— u) = — am u, sn (— u) = — sn u, 
 
 en (— u) = en u, dn (— u) — dn m. 
 
 705. sn2« + cn2« = l. 
 
 706. dn2« + /.-2sn2?f = 1. 
 
 707. dn^it - k-' cn'u = 1 - k' = k'^ 
 2 sn ?^ • en tt ■ dn ?( 
 
 708. sn2u = 
 
 709. en 2 M = 
 
 1 — k' sn'* « 
 
 cn^ i( — sn^?/ • dn^ u _ 1 — 2 sii^ ?t + k^ sn*u 
 1 - A;2 sn* ?< ~ 1 - A;2 sn* le 
 
 _ 2 sn^ ?< ■ dn^ u _ 2 cn^ ?< 
 
 1 — k^ sn* u 1 — k^ sn* u 
 
 „, » - _ dn^ ri — k' su'^ u ■ cn'^ u 1—2 k^ sn^ ?i + A;^ sn* u 
 
 710. dn 2 w = :j — J = — -^^ 
 
 1 ~ k- sn* i< 1 — A;'' sn* u 
 
 _ . 2 l^ ^v?u-Q,v?u __ 2 dn^ u 
 
 1 — k^ sn* M 1 — k^ sn* ?< 
 
 m\ 1 — en ?/ 1 — dn II. dn ii — en m 
 
 711. sn2( 
 
 712. cn^l 
 
 2 J l+dn« k\l + cnu) k'^ + dn u— k^ en u 
 n 4- en u k^ en v — k''^ + dn u 
 
 'u\ _ dn 
 
 + dn u k^(l + en n) 
 
 7c'- (1 +enM) 
 
 A;'^ + dn w. — k^ en w 
 
ELLIPTIC FUNCTIONS. 
 
 713. dirl-1- i + dui* ~ /l-2(l + cni*) 
 
 85 
 
 ~~ /v'^ 4- dn « — ^'■^ en u 
 
 If, moreover, v = I — , = > 
 
 Jo V(l - z') (1 - /tV) 
 
 714. sn^ u — sn^ y = cn^ v — cn^ ii. 
 
 „.. ^ ^ sn ?6 ■ en V • dn v =h en m • sn y • dn w 
 
 715. sn ill ±v) — —. :-, 5 
 
 ^ ^ 1 — k^ sn- u ■ sn'' v 
 
 „, _ , ^ en u • en y zp sn ?( • sn v ■ dn ?? • dn v 
 
 716. en (u ±v) = 7^ — 5 i 
 
 ^ ^ 1 — A;'' sn- tc • sn'' y 
 
 = en it • en V rp sn u • sn y • dn (u ± v). 
 
 „,„ , . . dn «-dn y =p />;'^ sn ?i • sn y? -en 7/ • en V 
 
 717. dn {u ±v) = ::; —„ 7, s 
 
 ^ ^ 1 — k^ sn^ u ■ sn'' v 
 
 = dn ?< • dn v zp /*;- sn ?< • sn ?; • en (w ± v). 
 
 „,„ , , tn ?f -dn f ± tn w-dn it 
 
 718. tn Cu±v) = - 7 , ^ 
 
 ^ ^ 1 ^tu u -tn V • dn u ■ an v 
 
 «,« , , s . V 2 sn it-en vdn V 
 
 719. sn (w + v) -\- sn (?t — v)= r; — 5 5— * 
 
 ^ "^ ^ ^ 1 — k^ sn'' « • sn'' v 
 
 / N 2 sn ?' • en u ■ dn ?t 
 
 720. sn (71 + y;) — sn (u — v) = r^ — 7. r~ ' 
 
 ^ ^ ^ -^ 1 — /v'' sn'' ?t • sn'' V 
 
 . . 2 en it • en ?' 
 
 cn(?t + w) + cn(it— y)= :j 1-5 3 ^• 
 
 ^ ^ ^ ^ 1 — /r sn- u ■ sn'' v 
 
 „„ _ ^ . , . 2 sn ti, ■ sn ?> • dn u ■ dn i; 
 
 Tad. en (it + iM — en ni ~ v)= 77 — 7, ^ 
 
 ^ ^ ^ ^ 1 — A:'' sn'' it • sn'' v 
 
 _,-„ ■, / ^ 1 / N 2 dn ?t • dn i> 
 7^ J. dn (u -\- v)+ dn (it — v)= 7-; ; ^ • 
 
86 ELLIPTIC FUNCTIONS. 
 
 _^ . , , , , , , 2k^ smi-snv -cnu-cnv 
 724. dn (ic + w) — dn (71 — v)= .. _ , .^ 
 
 sn^ u ■ sn^ V 
 
 sn^ u — sn'^ V 
 
 725. sn(t. + ^).sn(^^-^;) = ^_^,^^,^^_^^, 
 
 -y 
 
 1 — A:^ sn^ w • sn* v 
 
 1 Fdn^ V -j- k'^ sn^ ?* • cn^ v ~| 
 A;^ |_ 1 — A;^ sn^ u • sn^ v J 
 
 726. en (u ■}- v) ■ en (w — v) = rr — i ^ 
 
 en^ w + en^ v 1 _ -1 ^^^ ''^ ' ^^^ ^' + ^'^^ '^ ' ^^^^ ^ 
 
 1 — A;^ sn^ tc-su^v 1 — A;'* sn^ w • sn'' v 
 
 727. dn (u + v)- dn (m — v) 
 
 _ 1 — A;^ sn^ ?i — A;'^ sn^ v -\- k^ sn^ ii ■ sn^ v 
 1 — k^ sn^ u ■ sn^ t; 
 
 dn^ n + dn^ t? 
 
 -1. 
 
 
 1 
 
 -A:^ 
 
 sn^ 
 
 tt • sn^ V 
 
 
 sn?< 
 
 • dn w 
 
 • en V ± 
 
 sn ?' • dn V • 
 
 en u 
 
 
 1 
 
 -A;^ 
 
 sn^ 
 
 u • sn^ V 
 
 
 en?*- 
 
 dnw- 
 
 cnz?- 
 
 dn 
 
 vqzk'^ snu 
 
 • snv 
 
 1 — A;^ sn^ u • sn^ v 
 
 „_ _ , ^ , . sn M • en w • dn v ± sn ?? ■ en v • dn w 
 
 728. sn (u ± v) on {u rp v) 
 
 729. sn (m db v) dn (w qr v) 
 
 730. en(^±z;)dn(^zF^) = -^^ ^ j^ — sn^'t^^sn^.;^ 
 -n- .-^ , . x-ir^ , x-i (en V ± snw-dn w)^ 
 
 732. sn (wt, A;) = i sn (w, A;') /en (m, A:'). 
 
 733. en (wt, A;) = 1 /en (m, A;'). 
 
 734. dn {ui, k) = dn (m, k')/cn(it, k'). 
 
bessel's functions. 87 
 
 D. — Bessel's Functions. 
 
 7oO. t/g (x) — 1 2^ "^ 2^ • 4^ 2^ ■ 4^ • 6^ 
 
 736. iq, (a;) = ^0 (a;) • log a; + 2-2 - 2F:4r2 + 22.42.62 
 
 7i! A (- i)^-^.» + 2t 
 
 [a^.= l + i + i + ... + l/k:] 
 [When 11 is an integer 
 
 737 T I't\ — "V 5^ -^ [When ?i is an integ 
 
 • « I . t/„ ,^x; r (?i + 1) ^ 2" ■^'^^'■kl(n + k) ! ^19 may be used.] 
 
 738. lK(^)=Jn(x)-logx-^JX ^''~2^^.iy''' 
 
 
 
 
 
 739. According as n is or is not an integer, A ■ J^i^) + B ■ K„(x), 
 
 or A ■ J^(x) + B ■ J_ ,^(x) is a particular solution of Bessel's 
 
 equation, fp.^ ^ r?^ / w^ 
 
 + -■—+ 1--Az = 0. 
 
 740. f/Jg (a;) /riic = — J^ (cc) ; 6? [ic" ■ J"„ (x) ] /(7ic = x" ■J^_i (x) , 
 
 if /i > I ; (^[a;-« ■ J,Xx)ydx = — x.-" ■ J"„+i(a'), if ?i > - 1 
 
 741. J,^_,(x) - J„^,(x) = 2 . dj„(x)/dx ; 
 2 n ■ J^(x) = X ■ J"„_i (x) + x- J„+i(x). 
 
 When x is large it is sometimes convenient to compute 
 approximate numerical values of J^ (x) by means of the semi- 
 convergent series. 
 
 .^xrt x / N 2 r„ r (271 + 1)77 1 
 
 742. ^„(^)=^— I^P^.cosj^^ ^^_^| 
 
 (4 n? - 1) (4 n" - 9) 
 
 ^ . f(2n + l)7r 
 
 -.}]. 
 
 (4 rv" - 1) (4 ?i^ - 9) (4 v? - 25) (4 rv" - 49) 
 ^ 4 ! (8 .x)* 
 
 744 n - ^^'-^ _ (4^^-l)(4r.'^-9)(4n^-25) 
 
 '"~ 8x 3! (8^)^ "^ 
 
88 
 
 SERIES. 
 
 E. — Series and Products. 
 
 [The expression in brackets attached to an infinite series shows values 
 of the variable which lie within the interval of convergence. If a series 
 is convergent for all finite values of x, the expression [x^ < co] is used.] 
 
 745. (a + by = a" + na^'-'^b 
 
 , n(n — V) „,„ , , n\ a^~^b'' , ^,0^0-, 
 
 + 2! "''+•• • + (,.- A)! ,!.! +• -•■[*<"■] 
 
 746. (<i-Sx)-' = iri+- + ^ + ^' + ---1- pV<a».] 
 
 Ct I Cv ct- (M I 
 
 747. (1 zb xy = l±nx-\- ^ ^^~ ^ x^ 
 
 2! 
 n{n — V) {n — 2) x^ (± 1)^' n\ x^ 
 
 " 3] H • • • + ■^^^TT^yr^ + 
 
 748. (l±a;)-» = l=F?^^ + ^^^^^Vr^^' 
 
 2! 
 
 [a;2<l.] 
 
 3! ■ "^ ^^-^ {n-iy.k\ 
 
 749. (l±c.)i-l±ix-^.T^±|^a;^ 
 
 [x^ < 1, 
 
 J 
 
 
 750. (l±a,)-l = l=pj^+l^,x«^l^a^ 
 
 
 751. (l±.). = l±i.-y.'±if|^^ 
 
 1-2.5.8 , 
 3-6 -9 12 "■ 
 
 
 lx'<-L] 
 
SERIES. 
 
 89 
 
 1-4 , 1-4-7 , 
 752. (l±a:)-§ = l + ix + g^x2rF3-^a;' 
 
 1-4. 7-10 , r "^1 1 
 
 H 7i cc* zp • • •• \x- < 1.1 
 
 ^3 -6 -9 -12 ^ •- ^ 
 
 , a;* 1.3a;« 1.3-5ic« 
 
 753. (i±^¥ = i±i«^^-2:4±2:4:6~2T:6:8-'"- 
 
 [a;2<l.] 
 , 1-3 , 1.3-5 , , 
 
 754. (l±^')-^ = l + i^' + ^^ +27176 "^^ ■■■ 
 
 [x" < 1.] 
 
 755. (1 ± x)-' = l^x + x^-px'' + x*zpx'-\ . [a;2 < 1.] 
 
 31 , 311 3 
 
 756. (l±x)^ = l±ix-h^^x-:+j-^-^x' 
 
 3-1-1-3 3.1.1-3.5 ,<.^-j 
 
 + 2.4.6-8 ^2.4.6.8.10 ^ •- ^ 
 
 757. (l±^)-? = l + l^ + 2T4^'^t7I76^'"^"*' t^""'^^-^ 
 
 758. (I±x)-^ = 1^2x + 3x''zp4:x^ + 5x*^6x^+ ' • : 
 
 Ix' < 1.] 
 
 759. e- = l+x + |-' + |-j + ---. [a;=^<co.] 
 
 Cicloga)^ , (x log ay , r 2^.^ n 
 
 760. a- = 1 + X log a 4- '^ ^° ^ + ^—o, + ' " •• [^'<^^-] 
 
 761. i(«^ + e-^)=H-| + fi + fi + - •• [^^<^-] 
 
 762. i(e^-0 = a^ + fj + fj + f-! + '' ^- [x2<=^.] 
 
 763. «-" = l-^^ + fi-fi + li • I^^<oo.] 
 
90 SERIES. 
 
 A series of numbers, B^, B^, B^ •• •, of odd and even 
 orders, which appear in the developments of many functions, 
 may be computed by means of the equations, 
 
 2 n (2 n - 1) 
 
 2! 
 
 -^2n o, J^2n — 2 
 
 2n(27z-l)(2n-2)(2n-3) _ _ 
 
 —^ ^-B,„_,=(2n-1)B,,_, 
 
 _ (2.-l)(2.-2)(2.-3) ^^^_^^...^_^^„_,^^^ 
 
 Whence B, = h^2 = 1, B, = ^l, b, = 5,B, = ^\, B^ = 61, 
 B, = ^L, B, = 1385, B, = /^, Ao = 50521, Bn = ^Wo. ^12 = 
 2702765, Bis = i, etc. The ^'s of odd orders are called 
 Bernoulli's jSTumbers ; those of even orders, Euler's Numbers. 
 What are here denoted by -Z>2n-i ^nd Bz,, are sometimes rep- 
 resented by B„ and £J„, respectively, 
 
 7? 9 
 
 '2n — 1 
 
 (2?i)! (22«-l)7r2« 
 (2n 
 
 
 02n + 2r -I 1 1 1 
 
 7fi4 ^ _^ rr Ax^ B,x* B,x' B,x' 
 
 e^-1 2 2! 4! 6! 8! 
 
 [a;<2 7r.] 
 
 765. log X = (x - 1) - i(x - ly + i(x - ly . 
 
 [2>a;>0.] 
 
 
 [^>i-] 
 
SERIES. 91 
 
 [a^>0.] 
 
 768. log(l-{-x) = x-ix^ + ix'-ix* + -- -. [a;2<l.] 
 
 769. \ogCj^^ = 2[x + ix'-b^x' + \x'+-- 'I [rr2<l.] 
 
 771. log(a;+Vl+a;^) = a;-— + ^^j7g- ^^ g^ + -- •. 
 
 [a;2<l.] 
 
 Series for denary and other logarithms can be obtained 
 from the foregoing developments by aid of the equations, 
 
 log„a; = log, a: • log„e, log.x = log„a; • log, a, 
 log, (—z) = (2n + 1) iri + log,s. 
 
 772. sin £c = a; - |-j + |j - li H . [x^<co,] 
 
 jp2 ^4 ^6 
 
 773. cosa; = l— 77T + — — TTiH = 1 — versinic. [x^ < oo.] 
 
 2! 4! 6! 
 
 774. tan a; - a; + g + ^g + g^g + 2835 
 
 02n/02»i 1\ J> rp^n — l 
 
 + •••+ (2.)!"" +•••• [^^<i-^-] 
 
 ^py_ J^ «// JO ^ (/^ t// 
 
 775. ctna; = ----^- — -^^ 
 
 a;(2?i)! •- -■ 
 
92 SEEIES. 
 
 776. sec. = l + - + — +— +.- + -^^ + -.|_.^<-J 
 
 777. csc. = j + - + 3-g-, + — 
 
 ftmo • -1 i"'^ L • o X i. ■ o • O X 
 
 778. sm>a. = x + - + 2:^.- + 2:^.- 
 
 + ■ • • = ^ TT — cos~^a;. [a;^< 1.] 
 
 779. tan-^cc = x — ^x"^ + ^x^ - \x'' + • • • = ^17 — ctn-^x. 
 
 [a;2<l.] 
 
 780. tan-ix = ^-i + -^-p^3+-.-. [a;^>l.] 
 
 2 a; 3 a;'' 5ic^ ■- -* 
 
 „oi 1 ttI 1 1-3 1.3-5 
 
 781. sec~^a; = — — 
 
 2 a; 6a;' 2.4.5a;^ 2 • 4 • 6 • 7 a;' 
 
 = ^TT — csc"~^a;. [a;^>l.] 
 
 782. log sin a; = log a; - i a;^ - ^\^ x^ - ^^l^ a;« 
 
 22"-ii?,„_ia;2» 
 
 ?i 
 
 (2^0! 
 
 [a;2<7r2.] 
 
 783. log cos a; = - -^ a;2 - J^ x^ - J3. a;« - ^\\^ x» 
 
 027J-1 /92n _ -|\ 7? ^2n 
 
 ?i (2 /«) ! L * J 
 
 784. log tan a; = log x + ^ a;- + /^ *"* + ^§§5 ^^ 
 
 /92n-l _ -|\ 02« » „2n 
 
 w(2w)! L -i J 
 
 •yft*; Bin. 1 . ,^' Sa-" %x^ 2,x^ mx' ^ 
 
 [x2 < 00.] 
 
SERIES. 93 
 
 786. e-- = e(^l-- + — --g^+---J- [a:^<cc.] 
 
 787. e-^ = 1 + ^ + ^ + ^V ^' + ^ + • • •. [:.^<i7r^] 
 
 788. .sin-^ = l+:. + | + ^ + ^4----. [x^<l.] 
 
 789. e'^""'" = l+^ + |'-f -||-' • [^^<1.J 
 
 /yt'i /ytb /yti 
 
 *Aj tAj \Aj 
 
 790. sinha; = a; + - + gj + y^H . [a;=^<^.] 
 
 rg\^ /yt^ /yiO ™0 
 
 »A^ «Ay »A.' »Ay 
 
 791. coshx = l + - + - + - + -+---. [a;2<Q0.] 
 
 2! 4! 6! 8! 
 
 £c .^. ^. ^A ^ X 
 
 3 
 
 792. tanh x = (2' — 1) 2^^! — - (2* - 1) 2*^3 t] H 
 
 = :S[(-l)"-^22«(22»-l)^2„-ia;'"-V(2w)!]. 
 
 793. ctnh ic = - (1 + 2 [(- 1)"-' 22» j52„_i cc2V(2 ti) !]). 
 
 [a;2<7r2.] 
 
 794. sech cc = 1 + 2 [(- 1)» ^2„ ic'V(2 »0 H- [^' < i t^'-] 
 
 795. csch X = - - (2 - 1)2 B,^^ + (2^ - 1)2 B,^ 
 
 = i (1 + 2 2[(-l)"(2^"-^-l) ^2„_x 0^27(2 ^On)- 
 
 [a;2<7r2.] 
 
 796. sinh-^ = x-^x^ + ^^'-|^;|^ + --..[:«^<l.] 
 
94 
 
 SERIES. 
 
 797. tanh-ia; = x + | + ^ + ^+- • •. [a;^<l.] 
 
 798. ctnh-i ^ = --^T-s + ^5+-"' [a;2> 1.] 
 
 7QQ 1.-1 .1 1 , 1-3 1-3-5 
 
 7yy. cscn x-^ 2.3.x3'^2.4 Sx^ 2.4.6.7 •x'-''^ "'• 
 
 800. J|^V-'(Zic =a; - ^ x^ + ^ - -^^ + • • •. [a;2< QO.] 
 
 801. 
 
 n5 /yt9 /j^lo 
 
 J^'COS (X') <to = 0! - g^, + g^ - j^, + • ■ • . [^» < «.] 
 
 '•Xt 
 
 802. I ^— ^ = - ^ + 
 
 + x'' a a + b a + 2b a + 3b 
 
 + • • •. 
 
 803. f(x + h) = f(x) + h •/' (x + Oh). 
 
 h' 
 
 804. f(x + h) = f(x) + A ./' (X) + - /" (x) 
 
 805. /(x -h A) =f(x) -{-h-f'(x) + -/"(x) 
 
 nl ^ ' 
 
 + 
 
 A" 
 
 + (;^^^-^^~^^""'--^"^'^ + ^^> 
 
 806. /(x + A, y + A;) =/(x, li) + 7i/'^(a; + ^7i, y 4 ^A;) 
 
 + ^/'^(x + ^A, y + ^A;). 
 
 807. 
 
 
SERIES. 96 
 
 i (,/2ii^ + 3 « ^£(^ + 3 M' m^ 
 
 3!V ^^^ ^ ^Z/-^-^' ^-^-^y 
 
 +j,M^y...^E^ 
 
 ■.f(x, y) + QiD^ + ^^.)/(^, y) + I; {hD. + A-i),)y (a;, 2/) 
 
 ft • 
 
 ««« . 4 r . TTCc , , . Sttcc , , . Stto; n 
 
 808. 1 = - sm h i sin 1- i sm + • • • • 
 
 [0<ic<c.] 
 
 o«« 2 (■ r . Tra; , . 2 ttsc , , . 3 ttcc ~| 
 
 809. x = — sin A^ sm h ^ sm • ■ • 
 
 TT [_ c c c J 
 
 l-c<x<c.'] 
 
 c ^^-r TTiP , 1 3 7rar 1 ^irx ~| 
 
 810. ^ = ___^cos- + 3,cos— + ^cos-^ + ---J- 
 
 [0 < a- < c.-] 
 2'irx 
 
 «,-. . 2rr/7r2 4\ . irx tt^ . 
 811- -^=^LVT"l/"~~"2'^ 
 
 /tt^ 4\ . 37ra; tt^ . 4 
 + U-3-3>^"-^-4^^"- 
 
 + (—--, )sin — + ••• • [0 <»•<('.] 
 
 2 TTir . 1 3 irx 
 
 5 5V c 
 
 C^ 4 C^ r TTX 1 
 
 812. a;"-^ = r cos ^ cos h ^ cos 
 
 32 6- 
 
 42 — c 
 
 _l,cosi^ + ---]- l-c<x<c.-] 
 
m 
 
 SERIES AND PRODUCTS. 
 
 813 log sin ^x = — log 2 — cos x — ^ cos 2 a; — -^ cos 3x — • ■ '. 
 
 [0<a;<^7r.] 
 
 814. log cos ^x = — log 2 + cos x — ^ cos 2x + -^ cos 3 .r — • • • . 
 
 [0<ir<i7r.] 
 
 815. /(x) = i ^0 -I" ^1 cos — - -\- b^ cos (- • • • 
 
 . Tra; . 2 Tra- ^ ^ -, 
 
 + «! sin h agsm 1- • • •, f— c<.x<.c.\ 
 
 c c •- -" 
 
 where ^,„ = - I / (a) cos da, 
 
 1 r+'^ 
 
 C •>' — c 
 
 ?ft7ra , 
 sm da. 
 
 816. sin^=^ 1 
 
 [' 
 
 W,L 
 
 817. cos^ 
 
 .27r. 
 
 1-1 
 
 2^ 
 
 =[-(i')'] [-©)■] [-(.")•] 
 
 2^-4^6^ •• • (2m)^(2?>i + 2) TT 
 12.32.52 ... (^2 m + 1)-' 2 
 
 2^ -4' -6" • • • (2my(2m + l) 
 
 12.32.52 • • • (2m + ly 
 
 819. 
 
 ^"^^~2"n!l 2(2/^ + 2)^2- 
 
 a" 
 
 (2 /i + 2) 2 • 4 (2 ?? + 2) (2 7i + 4) 
 
 .r 
 
 2 • 4 ■ 6 (2 ;i -h 2) (2 w + 4) (2 ft + 6) 
 
 TT + 
 
 } 
 
820. 
 
 DERIVATIVES. 97 
 
 F. — Dekivatives. 
 
 d (au) a du 
 dx dx 
 
 R91 d{u + v) _du dv 
 dx dx dx 
 
 833. -^^; — - = V - — \- u--' 
 dx dx dx 
 
 fu\ du dv 
 
 QOQ \'V _ dx dx 
 
 dx v^ 
 
 824 ^/('^) = df(u) ^ du ^ 
 dx du dx 
 
 d\f{u) _ df (Ihi (lf_ dtl 
 dx^ du dx^ du? dx^ 
 
 826. ^ = ?^a;«-^ 
 dx 
 
 827. ^ = e-. 
 
 dx 
 
 oo« -^*" du . 
 
 829. '^^x^{l^\o^,x). 
 
 d{\o^^x) 1 log„e 
 
 830. 
 
 o?iC x • logg a a; 
 
 _-, c? sm a; 
 
 831. — ; = cos X. 
 
 dx 
 
 d cos X 
 
 833. — ; = — sm X. 
 
 dx 
 
98 DERIVATIVES. 
 
 833. — % = sec^ic. 
 
 ax 
 
 834. — = — csc^ic. 
 
 ax 
 
 835. — ' = tan x • sec x. 
 
 ax 
 
 836. — ; — = — ctn x ■ CSC x. 
 
 ax 
 
 ««-, (I sin~^a; 1 
 
 837. ; -- 
 
 ax 
 
 (I cos~^a:; 
 dx 
 
 d tan~^aj 
 dx 
 
 d etn^^.-B 
 dx 
 
 d sec~-^a; 
 dx 
 
 d csc~'rK 
 
 838. 
 
 839. 
 840. 
 841. 
 842. 
 
 Vl-x^ 
 
 -1 
 
 Vl-x« 
 
 1 
 
 1+x' 
 
 1 
 
 l+x2 
 
 1 
 
 x Vcc^ — 1 
 
 1 
 
 -.„ d^rahx . 
 
 843. ; = cosh a;. 
 
 dx 
 
 _. - (^ cosh a; . , " 
 
 844. — ^- — = smh x. 
 
 ax 
 
 o._ d tanh x . „ 
 
 845. ; = sech^ x. 
 
 dx 
 
 846. ^li^=-csch^... 
 
 dx 
 
DERIVATIVES. 99 
 
 847. = — seen x ■ tanh x. 
 
 ax 
 
 848. = — cscli X ■ ctnli x. 
 
 ax 
 
 _._ d sinh~^a:; 1 
 
 849. - 
 
 dx -yjx^ + 1 
 
 d cosh" ^ a; 1 
 
 850. 
 851. 
 852. 
 853. 
 854. 
 855. ^^£j{x)dx = f{h) 
 
 dx -^x'-l 
 
 d tanh~ ^x _ 1 
 dx 1 — x^ 
 
 d ctnh^^a:; _ 1 
 d sech~^a; — 1 
 
 dx X Vl — x^ 
 
 d csch~^a; — 1 
 
 dx X Va;"' + 1 
 
 d ^^ 
 
 db. 
 
 856. ^fy(x)dx = -f(a). 
 
 857. jjjix, c) dx =£l)J(x,c) . dx +f(b, c) g - f(a, c) ^• 
 
 r.,ro d«(u-v) d"u , dv d"-'^u 
 
 858. — ^^ = V ■ h n-- ; r 
 
 dx" dx" dx dx"-^ 
 
 n(n — V) d^v d"-^ic , d'^v 
 
 ^ 2! dx^ dx^-^ dx'' 
 
 859. If f(x, y, z, • • •) is a homogeneous function of the wth 
 order, so that /(Ax, Xy, \z, • • °) ^ X"/(x, y, z, • • •), 
 
 x-DJ+y.DJ+z.DJ+-'- = 7if. 
 
100 DERIVATIVES. 
 
 860. Ux = <f>(y), 
 
 dy _ 1 A _ _ < ^"(y) 
 
 15 
 
 861. If sc = f{t) and ?/ = (^ {t), 
 dy^£({) d'y_ f'(t)-<f>"(t)-f"(t).<f>'(t) 
 
 dx f'(ty dx' [f'(t)Y 
 
 862. Uf(x,y) = 0, 
 
 dy ^ V >^f_ DJ 
 dx dx ' dy D^f 
 
 dhi _ D,y • ipjY - 2 D^BJ. DJ. DJ+ D,y ■ jDJ) 
 dx' {DJ) 
 
 863. If y =f(u, v), u = <f>{x), and v = \p{x), 
 
 d£ ^^i dM di <ii^ 
 
 cZx gw dx ^ dv dx ^«/ + ^ ^"/' 
 
 ^_ay /^iA" ay du dv_ dy^ 
 
 dx^ du^ \dxj du ■ do dx dx d'' 
 
 df d?u df dh 
 du dx^ dv dx^ 
 
 ■v \dxj 
 
 = u'' ■ D\f +2u'-v'- L„ DJ+ v'^ ■ DJ^f 
 + u".I)J+v".DJ. 
 
 864. If f{x, y, z) = 0, D^--^- DJjDJ, 
 DJz = -lD:'f.{DJf 
 
 - 2 DJ. DJ- DJ)J^ D,y{DJy^J{DJf, 
 Djy^z = - ID^DJ- {Djy - DJDJ. D^DJ 
 
 + DJ. DJ. DJ)J+ DJ. DJ. DJ^Iipjy 
 
DERIVATIVES. 101 
 
 865. If r=</)(w, v), u=fx{x,y), and v=f^{x, u), 
 
 D^ V= d: <i> . (D,uy+ d: </, . {D^vf + 2 D^D, ^ . D,u ■ D,v 
 + 2 i>„D,«/> • {_D,u ■ D^v + D,jU ■ D.y] 
 
 In the special case, u^r = Va-'^ + y'^, v = 6 = tan~^ (^//a-), 
 we have D^x = cos = x/ Va;'^ + ?/- ; D^y = sin = y I Va;- + ?/^; 
 Z>0X = — r sin = — y \ D^y = r cos ^ = a; ; 
 
 D^r = a; / Va;^ + y'^ = cos ^ ; Z>^r = ?// •va^M-l? = sin ^j 
 
 I^:P=-y I (^' + 2/') = - sin ^/r- ; 
 Dyd=x / (x^ + y^) = C0Bd /r\ and 
 
 Z>/ F + X*,;- F = Z>,^ F + - -i), V+\- De" V. 
 
 866. If F= «^(?<, v)' ''*=/iO^ ^). and v = f^{r, 6), 
 
 2>,2 F + ^ • A- F + i ■ A' F = i)„2 F- [(Aw)' + ^^$^~\ 
 
 r r 
 
 
102 DERIVATIVES. 
 
 867. If V=4>{u, V, iv), u =fi{x, ij, z), V =f^{x, y, z), and 
 
 w=fs(x, y, z), 
 
 I)JV= BJ^V. (R,icy + D^'V. (D^vf + DJV- (D^wf 
 
 B^ V + D; V + D,'V= D^'V- [{D^uy + (!>,«)'+ (A^O'l 
 
 + i),^ r[(D,^(;)^ + (D^tvy + (D^wy^ 
 + 2 i)„A F- [ J>,tt . X>,^; + D,it ■ J>,v + JD^u ■ A«] 
 + 2 1),A,V- IB.v ■ D^w + D,^v . D,jw + D,v ■ B.w^ 
 + 2 i)„.i)„ V- [Bjv ■ JDji + D,/a ■ D^u + D^^v ■ i),w] 
 + D„F.[i)> + I>/« + A'^*] 
 
 In particular, if 
 
 a; = r sin 6 cos <^, y = ?' sin 6 sin ^, z = r cos ^, 
 30 that M = j-^^ = a;2 + y2 _^ ,-s^ i; - ^ = tan-^ ( Va;^ -f- f/z), 
 w~^~ tan~^ {y /x), we have 
 
 Z>,.« = cos ^ = s/ Vx^ + ?/ + .v^ ; 
 D^x = sin ^ cos ^ = x / VxM- y^ + ^^ ; 
 
DERIVATIVES. 
 
 103 
 
 r^ sin 6 
 
 V.y — sin 6 sin <^ = y I V.^'- + if + z^\ 
 
 I)qZ = — r sin ^ = — Vcc^ + if ; 
 
 jD^ = r eos ^ cos <^ = zx / ^j^ + y^ : 
 
 J)^y z= r cos ^ sin ^ = zi/ / Vo;- + 3/^ ; 
 
 B^z^O; 
 
 D^x = — r sin ^ sin ^ = — y ; 
 
 X)^y = r sin ^ cos <^ = x ^ 
 
 7)_r = s/r = cos 6\ 
 
 D£ = - Vx-2 + //r^ = - sin 0/r j 
 
 Z)^?- = X /r = sin ^ cos <^ ; 
 
 J)^0 = xz/7-^ 'Vx^ -{- f = COS cos <;?>/r; 
 
 ^;«<^= -I//(^^ + f)= -sin<^/r sin^j 
 
 -^/ = y/-'" = sin ^ sin <^ ; 
 
 Z)y$ = ^1/ / i^ Va;^ + y"^ — cos ^ sin ^/r; 
 
 X>j,<^ = X / (x^ + U') = cos ^ /r sin 6 ; 
 
 (i>,r)^ + (i>,r)^ + (D^rf = 1 ; 
 
 (DAY + (A,<^)^ + (A<^)' - 1 /'-^ siii'^ ; 
 
 D^^V + I),fV + B.^V 
 
 D,{r- ■ B, V) ■sinO + ^^ + A(sin 6 ■ A F) 
 
104 
 
 DERIVATIVES. 
 
 868. If X =fi{u, v), y =/2(m, v), z =fz{u, v), 
 
 D^ = 
 
 _ D..f,-Dj\~DJ,-D,.f, 
 
 869. If X =/(«, u), and y = ^(z, u), 
 
 Byz = BJlip,^. BJ- BJ. i>„<^). 
 
 870. If F^ (x, y, z, u, v) = 0, 
 
 F^ (x, y, z, u, v) — 0, and F^ (x, y, z, u, v) = 0, 
 
 B^ 
 
 B,F, B^F, B^F, 
 B^F, B^F, B,F, 
 B,F, B^F, B,Fz 
 
 B,F, B^F, B^F, 
 B,F, B^F, B,F, 
 B,F, B^F, B,F, 
 
 871. If F^ (x, y, z) = 0, and F^ (x, y, z) = 0, 
 
 cly dz 
 
 B,F, . B,F^ - B,F, . B,F, BJ\ • B,^F, - B,F^ ■ B,,F, 
 
 dx 
 ByF^.B,F^-B^F^-B,F,' 
 
 If each of the quantities y^, y^, yz, • • • 2/„ is a function of 
 the n variables x^i x^, x^, ' • • x^, the determinant, 
 
 B^^yi B^^j^ B^^y^ • • • 
 B^^y^ B^^j^ B^^j^ ■ ' • 
 
 B^^y„ B^,^y„ D^ij,^ ■ • • B^j„ 
 
873. 
 
 DERIVATIVES. 105 
 
 is called the functional determinant or the Jacobian of the 
 ^s with respect to the a;'s and is denoted by the expression, 
 
 g,jr2 g(yi> y2> 2/3, •• • Vn) . g (^^^1, ^2, X^, • ' ' X„) ^ ^ 
 d {Xi, a-2, Xs, ' • ■ Xn) d (3/1, y2, Vz, • • • Vn) ~ 
 
 d (Vl, ?/2, Vs, ■ ■ • Vn) . g (^1, ^2, ^3, • • • ^„) 
 g (^Ij ^2) ^3j ■ ■ ' ^n) g (^1? "^25 X^, • • • X^ 
 
 ^ d (]/l, 3/2, y?., ■ • • Vn) 
 (Xi, X^f Xg, • • • X^) 
 
 If the ?/'s are not all independent but are connected by an 
 equation of the form (f> (jji, ?/2, ys, ' ' ■ y„) = 0, the Jacobian 
 of the ?/'s with respect to the cc's vanishes identically ; and, 
 conversely, if the Jacobian vanishes identically, the ?/'s are 
 connected by one or more relations of the above-mentioned 
 form. 
 
 The directional derivative of any scalar point function, u, 
 at any point, P, in any fixed direction PQ\ is the limit, as 
 PQ approaches zero, of the ratio of «q — Up to PQ, where 
 ^ is a point on the straight line PQ' between P and ^'. The 
 gradie7it, h^, of the function ti at P is the directional deriva- 
 tive of M at P taken in the direction in which w increases 
 most rapidly. This direction is normal to the surface of 
 constant m which passes through P. 
 
 874. K' = {D,u)' + {D^n)' + {D^uf. 
 
 The directional derivative of any scalar point function at 
 any point in any given direction is evidently equal to the 
 product of the gradient and the cosine of the angle between 
 the given direction and that in which the function increases 
 most rapidly. 
 
106 MISCELLANEOUS FORMULAS. 
 
 The normal derivative, at any point, P, of a point function 
 u, taken with respect to another point function v, is the limit 
 as P(^ approaches zero of the ratio of «q — tip to Vq — Vp, 
 where ^ is a point so chosen on the normal at P of the 
 surface of constant v which passes through P, that Vq — Vp 
 is positive. If (u, v) denotes the angle between the directions 
 in which u and v increase most rapidly, the normal derivatives 
 of u with respect to v, and of v with respect to u may be 
 written 
 
 h^^ cos (?<, v) -7- 7ij,, and A„ • cos (w, v) -v- ^„ 
 
 respectively. If A„ = h^, these derivatives are equal. 
 
 Gr. — Miscellaneous Formulas. 
 
 If s is a plane analytic closed curve, n its normal drawn 
 from within outwards, and dA the element of plane area 
 within s, the usual integral transformation formulas for the 
 functions u and v which, with their derivatives of the first 
 order, are continuous everywhere within s, may be written — 
 
 875. I M • cos (x, n) ds = \ i D^u ■ dA. 
 
 876. j [w • cos (x, 7i) + V ■ cos (?/, w)] ds=^ C C(D^ti + DyV) dA. 
 Sn. Jb„u .ds= C C (B/u + Dyhi) dA. 
 
 878. j'^iD.n . Djj + D,^u ■ D^v) dA 
 
 = Ju ■ D^v -ds- C Cu (Z)> + D,fv) dA 
 = Cv . D^u .ds-^ifv {B^u + D,fv) dA. 
 
 879. f C\ {D^u ■ Djo + D,;ii ■ IJ,/-) dA = Cxu- D„v ■ ds 
 
 -ff ' U'. (^ • A'O + ^. (^ • ^>M dA 
 
MISCELLANEOUS FORMULAS. 107 
 
 If ^ and 7} are two analytic functions which define a set of 
 orthogonal curvilinear coordinates, and if (^, n) and {-q, n) 
 represent the angles between n and the directions in which 
 ^ and 7], respectively, increase most rapidly. 
 
 880. ^j'h^ ■ \ • A ( r ) ^^ =X" ■ ^°^ ^'^' ^^^ ^^' 
 
 881. ^ ^ h^ h^-DA^jdA =fu . cos (i, n) ds. 
 
 882. If r is the distance from a fixed point, Q, in the coordi- 
 nate plane, 
 
 /cos (v 71) cl'S 
 — '-^^^ — —— = 0, TT, or 2 TT, according as Q is without, 
 
 on, or within s. 
 
 If a9 is an analytic closed surface, n its normal drawn from 
 within outwards, and dr the element of volume shut in by S, 
 the usual integral transformation formulas may be written — 
 
 883. r Cu cos (x, 71) dS= C C C D^u ■ dr. 
 
 884. I I [y« cos {x, n) + v cos (y, n) + tv cos (z, n)"] dS 
 
 = f f fi^x'if' + I>y^ + D,w)dT. 
 
 885. r rz)„« • (7s = ( ( r (^/« + ^2,''* + a'«) <^t. 
 
 886. j" ^ j" (7),^« . D^v + i>^7f . D^v + i),-a . X>,y) dr 
 
 = r r« -D^v-dS- C C Cu (Bj'v + Z>/y + n^^) dr 
 = f f^- ^nU dS- C C C r {Dju + Dfa + D.hi) dr. 
 
108 MISCELLANEOUS FOKMULAS. 
 
 887. fff>^ {D^u ■ D,v + DyU ■ D^v + D,u ■ D,v) dr 
 
 - ^ ^ ^ vlB^iXB^u) + D^iXD^u) + D,{XD,u)^dr, 
 
 If I, rj, I are three analytic functions which define a system 
 of orthogonal curvilinear coordinates, 
 
 889. jyj"/'| • ^ • h^ ■ Dr, (j;^) ^^ =ff"' ■ cos (^7, ?0 ^'S^- 
 
 890. ////^f • hr, .\-D^ {irrh) '^^ =//" • ^°' (^' '') '^'^- 
 
 891. If r is the distance from a fixed point, Q, 
 
 /cos ^?' ??'^ 
 -j—^ dS = 0, 2 TT, or 4 TT according as Q is without, 
 
 on, or within S. 
 
 Stokes's Theorem, — The line integral, taken around a 
 closed curve, of the tangential component of a vector point 
 function, is equal to the surface integral, taken over a surface 
 bounded by the curve, of the normal component of the curl of 
 the vector, the direction of integration around the curve form- 
 ing a right-handed screw rotation about the normals. 
 
 If X, Y, Z are the components of the vector, 
 
 892. C{Xdx + Ydy + Zdz) = C C[(D,,Z - I), Y) cos (x, n) 
 
 + (I),X ~ D,Z) cos (t/, n) 
 
 + (B, Y - DyX) cos {z, n)-] dS. 
 
MISCELLANEOUS FORMULAS. 109 
 
 Equations 893 to 897 give Poisson's Equation in orthogonal 
 Cartesian, in cylindrical, in spherical, and in orthogonal curvi 
 linear coordinates. 
 
 893. v2r=Z»/r+X>/F+ A'^=-4 7rp. 
 
 1 
 
 894. ~I),(r.D,V) + ^-De'r+l),^V=-4.7rp. 
 
 895. sme.DJr^-D,.V) + ^^ 
 
 ^ ^ sm 6 
 
 + Dg (sin ei)0V) = - A Trpr'' sin 6. 
 
 896. 7i/ ■D^^V+ /i,2 ■D^W+ hi ■ Dl V 
 
 397. ;,,./,,. /.,{i),(,^^^.A^')+ A 
 
 _rL 
 
 •i>„F 
 
 y.,A, ^ 
 
 H. — Certain Constants. 
 
 7r = 3.14159 26535 89793 
 logio7r = 0.49714 98726 94134 
 
 - = 0.31830 98861 83791 
 
 TT 
 
 TT^ = 9.86960 44010 89359 
 
 V^ = 1.77245 38509 05516 
 
 logio 2 = 0.30102 99956 63981 
 
 e = 2.71828 18284 59045 
 
 logio e = 0.43429 44819 03252 
 
 log, 10 = 2.30258 50929 94046 
 
 log^2 = 0.69314 71805 59945 
 
 togiologio e = 9.63778 43113 00537 
 
 log^7r = 1.14472 98858 49400 
 
 + ^dj±-^<r)^ = -*^P 
 
110 FORMULAS OF INTEGEATION. 
 
 I. — General Fokmulas of Integration. 
 
 F and / represent functions of x, and F\ /', F'\ f", their 
 first and second derivatives with respect to x. 
 
 898 
 899 
 900 
 901, 
 
 Cf' •/• dx = Ff- Cf-/' ■ dx. 
 
 . C{Fy-F'-dx = (Fy+^/(u + i). 
 
 C(aF + by ■ F' ■ dx = (aF + by + ^/a (n + 1\ 
 
 J(F +fy -dx ^Jf(f +fy-'dx +Jf{F +f)-'dx. 
 
 902. CF'/(Fy-dx = -l/(n-l)(Fy-\ Cf'/F- dx = logF. 
 
 903. /(^'-Z- F.f ')/(/)''. dx = F/f. 
 
 . f{F'-f- F-f)IFf. dx = log {FIf). 
 
 J dx _ _1^ r dx d_ r dx 
 
 F-{x^ -a?) ~2^,J F-{x-a) ~ 2~aJ F ■ {x + a) 
 
 r dx _ r dx r dx 
 
 ■JF(F±f)~' J F.f^J f{F±f)' 
 
 /pi fly, 
 
 , = (2^aF+b)/a. 
 
 y/aF + b 
 
 C F'-dx , ,„ /-— , 
 
 I , = log {F + -^I^' + a). 
 
 / Fdx _ a r dx b f dx 
 
 {F+a)(F + b) ~ a-bJ Y+~a. ~ a-bJ F+b 
 
 r F-dx _ r dx r fdx 
 
 J (F+fY~ J (F+ f)"-' ^ J TF 
 
 910 
 
 {F+fy j(F+fy-^ ^{F+fy 
 
 911 r_Zji^ = l -I'lZ r F'-dx ^ 1 ^ qF-p 
 
FOEMULAS OF INTEGllATION. Ill 
 
 913. f f! '"". =- tan-', 
 J F^ -\- a} a \ a 
 
 F'-dx 1. ^lF\ 
 
 aF — b 
 
 aF +b 
 
 914. I -—- — T7, = TTT ^°S 
 
 rF^^dx _ r F^-dx r t^"-. 
 
 or./; 
 
 {F'^ -h) V 2(i'^»+^' 
 
 
 F'^d^__ 1 
 
 + hF~ h '"^ aF +b 
 
 917. " .. . , „ = T loR- 
 
 
 919. f , ^' = y sec- ' ( ^ 
 
 r 
 
 J — INTEGRALS Useful in the Theory of Alternating 
 
 Currents. 
 
 922 
 923 
 
 924 
 
 I sill ((nt -{- (f>)dt = • cos (wt + </)). 
 
 . I cos (oit -\- ^)dt = -- sin (oit -\- <(>). 
 
 /I 1 
 
 sm^(u)t + <f)dt^-t — -— sin 2 (i^t + </>). 
 
112 AUXLLIAIIY FORMULAS. 
 
 925. / sin (uyf + <^) ■ cos (wf + cl>)dt = — - sm'^(wt + <}>). 
 
 /I 1 
 
 cos2(o)^ -\- <t>)dt = -t + -— sin 2 (u)t + cfi). 
 
 927. fsin (.ot -\- X) ■ sin (a>^ + /x) dt = ^^^ — ^ (o)^) 
 
 sin (<i)t + A) • cos (wt + /^) 
 2 (1) 
 
 _„_ I . , , , ^ , sin (<Dt + A) • sin (wt -\- a) 
 928. / sin (oyt + A^ • cos (oyt 4- m) dt = ^^ — ^^-^ ^^ — ^^-^ 
 
 / sin ((at + A) • cos (wt 4- /u.) f/i^ 
 
 sin (/Lt — A) 
 
 (-0- 
 
 929. / cos (wt + A) • cos (wt + /x) fZi = ^"^ ^^ ^ (w^) 
 
 sin (<«>^ + A) • cos (wt + A) 
 2 w 
 
 ^„^ C . , . , ^ , sin r???f — n?' + A — Atl 
 
 930. / sm (»^ ^ -I- A) • sm (ni + yu) </^ = ^-—-^ =4 ^ 
 
 J 2 (?« — n) 
 
 sin [w/ + ?t^ 4- A + /"■] 
 2 (m + w) 
 
 931. f cos (mt + A) • cos int + ix) dt = — ~- ; — ; 
 
 J ' 2 (/?» + n) 
 
 sm\^nit — nt -{- X — fi'] 
 2 (in — ri) 
 
 o,«« r ■ , s , s , cos r??2i + nt + X+ ul} 
 
 932. / sin r^/^i + A) • cos ^nr + n) dt = ^-— 9= , ^ ^-' 
 
 2 (m + ?'') 
 
 cos [?wi — nt -\- X — fJi] 
 
 2 (»i - n) 
 
 J. I sin (?//i -}- A) • cos (ni + fx) dt = 
 
AUXILIARY FOIlxMULAS. 
 
 113 
 
 933 
 
 '. I cos ((tit + A + vi.r) ■ COS (oj^ + A — III J') dx 
 = cos'^(w^ + A) 
 
 inx -\- sin m.r • cos iiix 
 mx — sin mx • cos inx 
 
 2«i. 
 
 -sin2(w;' + A) 
 
 m- sin(w/'4-<^)4-?i- cos(w)'+<^) = V»r+H.2- sin(a)f+<^+c)^ 
 
 \ where tan c = n/m. 
 
 III ■ sin(<Df+^)— w • cos(w!'+<^)= V//r+?i-^ • sm{(t)t-\-<i>—d). 
 
 934. 
 
 / 
 
 e^-^^"^'(h^ 
 
 -h^ cl 
 W + c^ 
 
 ^e 
 
 .(-b±ci)t 
 
 -ht 
 
 = 77 ; [(c ■ sin ct — b- cos ct) =F «' G> ■ sin r?' 4- c • cos cf)! 
 
 0^ 4-c^^ 
 
 r.-ht 
 
 —= [sin (cf - 8) =F '■ • cos (c;^ - 8)] 
 
 935. j e"' ■ cos (<of + tf>) (If 
 
 a^ + o, 
 
 where tan S = /^/c. 
 
 ^ [to sin ((of + ^) 4- « • cos ((of + <^)] 
 
 = — -==:^ cos [to;' + </) — tan ^(a)/«)]. 
 936. C (>"'■ sin ((of + <f>) (It 
 
 ^ai 
 
 a^+(o 
 
 2 [a • sin (wf + ^) — o> • cos ((of + <^ )] 
 
 937. /^[e''' • sin ((of + <i>)'fdt 
 
 ■ sin \_(of + <^ — tan ^(w/a)]. 
 
 4 
 
 1 to • sin 2 ( o)/' 4- <^) + « ■ cos 2 (ojj' 
 
 '^ + <^) ] 
 
 a cf'^ + ft)"^ J 
 
 1 cosr2a)?' + 2(^-tan-i(ai/Q:)] " 
 
 (X 
 
 Va"^ + 
 
 U) 
 
 '1 
 
114 
 
 AUXILIARY FORMULAS. 
 
 938. fie"' ■ COS (wt + <i»)ydt 
 
 „2at 
 
 1 o) • sin 2 (u)^ + ^) + (T • cos 2 (wt + ^) " 
 
 a 
 
 « 
 
 «-^ + 
 
 1 cos[2W + 2<^-taii-Xco/«)] "| 
 
 V^7+^ 
 
 In the case of a direct trigonometric function of (wt + <!>), 
 T = 2 tt/o) is called the ^>erio(^ or the c//f^e. The mean 
 value for any whole number of periods, reckoned from any 
 epoch, of sin (wt + <^), cos (wt + (j>), or sin (cof + ^) • cos (wf + </>), 
 is zero, whereas the mean value for any whole number of half 
 periods, reckoned from any epoch, of either sin'-^ (wt + <f)) or 
 cos^((ot + (f>) is one half. The mean value of sin (w.') from 
 ^ = to ;* = ^ T, or of cos (<ot) from - ^ T to + i T, is 2/7r 
 or 0.6366. 
 
 The mean value, for any number of whole periods, of either 
 sin(a)?'+X) • sin(co^+/x) or cos(cu?'+X) • <-os(wf+fi) is h ■ cos(A— /x), 
 while the mean value of sin(wi; + A) ■ cos ((Dt -\- /x) is ^ sin (\ — fi). 
 
TABLES. 115 
 
 INTERPOLATION. 
 
 If values of an analytic f unction, /(x), are given in a table for a number 
 of values of the argument x, separated from one another consecutively by 
 the constant small interval, 5, the differences between successive tabular 
 values of the function are called ^rsi tabular differences, the differences of 
 these first differences, second tabular differences, and so on. The tabular 
 differences of the first, second, third, and fourth orders corresponding to 
 z= a are 
 
 Ai=/(a + 5)-/(a), 
 
 A2 =/(a + 2 5) - 2 -/(a + 5) +/(a), 
 
 A3 =f{a + 3 5) - 3 -/{a + 2 5) + 3 -/{a + 0) -f{a), 
 
 A4 =/(a + 4 5) - 4 .f{a + 3 5) + 6 -/(a + 2 5) - 4 ./(a + 5) +/(a), 
 
 where / (a) is any tabulated value. 
 
 The value of the function for x = {a + h), v?here h = kS, is 
 
 Z.X ^. V , * k{k-l) ^ k(k-l)(k-2) ^ 
 f{a + h) =/(a) + A; • Ai + ^^^ • A2 + ^ ^ '- ■ A3 
 
 _^ Mfc-l)(fc-2)(A:-3) _^_.^^^^ 
 
116 
 
 TABLES. 
 
 "he Probability Integral. 
 
 
 
 
 1 
 
 ( 2 
 
 '0 
 
 dx. 
 ) 
 
 
 
 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 0.00 
 
 0.00000 
 
 00113 
 
 00226 
 
 00339 
 
 00451 
 
 00564 
 
 00677 
 
 00790 
 
 00903 
 
 01016 
 
 0.01 
 
 0.01128 
 
 01241 
 
 01354 
 
 01467 
 
 01580 
 
 01792 
 
 01805 
 
 01918 
 
 02031 
 
 02144 
 
 0.02 
 
 0.02256 
 
 02369 
 
 02482 
 
 02595 
 
 02708 
 
 02820 
 
 02933 
 
 03046 
 
 03159 
 
 03271 
 
 0.03 
 
 0.03384 
 
 03497 
 
 03610 
 
 03722 
 
 03835 
 
 03948 
 
 04060 
 
 04173 
 
 04286 
 
 04398 
 
 0.04 
 
 0.04511 
 
 04624 
 
 04736 
 
 04849 
 
 04962 
 
 05074 
 
 05187 
 
 05299 
 
 05412 
 
 05525 
 
 0.05 
 
 0.05637 
 
 05750 
 
 05862 
 
 05975 
 
 06087 
 
 06200 
 
 06312 
 
 06425 
 
 06537 
 
 06650 
 
 0.06 
 
 0.06762 
 
 06875 
 
 06987 
 
 07099 
 
 07212 
 
 07324 
 
 07437 
 
 07549 
 
 07661 
 
 07773 
 
 0.07 
 
 0.07SS6 
 
 07998 
 
 08110 
 
 08223 
 
 08335 
 
 08447 
 
 08559 
 
 08671 
 
 08784 
 
 08896 
 
 0.08 
 
 0.09008 
 
 09120 
 
 09232 
 
 09344 
 
 09456 
 
 09568 
 
 09680 
 
 09792 
 
 09904 
 
 10016 
 
 0.09 
 
 0.10128 
 
 10240 
 
 10352 
 
 10464 
 
 10576 
 
 10687 
 
 10799 
 
 10911 
 
 11023 
 
 11135 
 
 0.10 
 
 0.11246 
 
 11358 
 
 11470 
 
 11581 
 
 11693 
 
 11S05 
 
 11916 
 
 12028 
 
 12139 
 
 12251 
 
 0.11 
 
 0.12362 
 
 12474 
 
 12585 
 
 12697 
 
 12808 
 
 12919 
 
 13031 
 
 13142 
 
 13253 
 
 13365 
 
 0.12 
 
 0.13476 
 
 135S7 
 
 13698 
 
 13809 
 
 13921 
 
 14032 
 
 14143 
 
 14254 
 
 14365 
 
 14476 
 
 0.13 
 
 0.14587 
 
 14698 
 
 14809 
 
 14919 
 
 15030 
 
 15141 
 
 15252 
 
 15363 
 
 15473 
 
 15584 
 
 0.14 
 
 0.15695 
 
 15805 
 
 15916 
 
 16027 
 
 16137 
 
 16248 
 
 16358 
 
 16468 
 
 16579 
 
 16689 
 
 0.15 
 
 0.16800 
 
 16910 
 
 17020 
 
 17130 
 
 17241 
 
 17351 
 
 17461 
 
 17571 
 
 17681 
 
 17791 
 
 0.16 
 
 0.17901 
 
 18011 
 
 18121 
 
 18231 
 
 18341 
 
 18451 
 
 18560 
 
 18670 
 
 18780 
 
 18890 
 
 0.17 
 
 0.18999 
 
 19109 
 
 19218 
 
 19328 
 
 19437 
 
 19547 
 
 19656 
 
 19766 
 
 19875 
 
 19984 
 
 0.18 
 
 0.20094 
 
 20203 
 
 20312 
 
 20421 
 
 20530 
 
 20639 
 
 20748 
 
 20857 
 
 20966 
 
 21075 
 
 0.19 
 
 0.21184 
 
 21293 
 
 21402 
 
 21510 
 
 21619 
 
 21728 
 
 21836 
 
 21945 
 
 22053 
 
 22162 
 
 0.20 
 
 0.22270 
 
 22379 
 
 22487 
 
 22595 
 
 22704 
 
 22812 
 
 22920 
 
 23028 
 
 23136 
 
 23244 
 
 0.21 
 
 0.23352 
 
 23460 
 
 23568 
 
 23676 
 
 23784 
 
 23891 
 
 23999 
 
 24107 
 
 24214 
 
 24322 
 
 0.22 
 
 0.24430 
 
 24537 
 
 24645 
 
 24752 
 
 24859 
 
 24967 
 
 25074 
 
 25181 
 
 25288 
 
 25395 
 
 0.23 
 
 0.25502 
 
 25609 
 
 25716 
 
 25823 
 
 25930 
 
 26037 
 
 26144 
 
 26250 
 
 26357 
 
 26463 
 
 0.24 
 
 0.26570 
 
 26677 
 
 26783 
 
 26889 
 
 26996 
 
 27102 
 
 27208 
 
 27314 
 
 27421 
 
 27527 
 
 0.25 
 
 0.27633 
 
 27739 
 
 27845 
 
 27950 
 
 28056 
 
 28162 
 
 28268 
 
 28373 
 
 28479 
 
 28584 
 
 0.26 
 
 0.28690 
 
 28795 
 
 28901 
 
 29006 
 
 29111 
 
 29217 
 
 29322 
 
 29427 
 
 29532 
 
 29637 
 
 0.27 
 
 0.29742 
 
 29847 
 
 29952 
 
 30056 
 
 30161 
 
 30266 
 
 30370 
 
 30475 
 
 30579 
 
 30684 
 
 0.28 
 
 0.30788 
 
 30892 
 
 30997 
 
 31101 
 
 31205 
 
 31309 
 
 31413 
 
 31517 
 
 31621 
 
 31725 
 
 0.29 
 
 0.31828 
 
 31922 
 
 32036 
 
 32139 
 
 32243 
 
 32346 
 
 32450 
 
 32553 
 
 32656 
 
 32760 
 
 0.30 
 
 0.32863 
 
 32966 
 
 33069 
 
 33172 
 
 33275 
 
 33378 
 
 33480 
 
 33583 
 
 33686 
 
 33788 
 
 0.31 
 
 0.33891 
 
 33993 
 
 34096 
 
 34198 
 
 34300 
 
 34403 
 
 34505 
 
 34607 
 
 34709 
 
 34811 
 
 0.32 
 
 0.34913 
 
 35014 
 
 35116 
 
 35218 
 
 35319 
 
 35421 
 
 35523 
 
 35624 
 
 35725 
 
 35827 
 
 0.33 
 
 0.35928 
 
 36029 
 
 36130 
 
 36231 
 
 36332 
 
 36433 
 
 36534 
 
 36635 
 
 36735 
 
 36836 
 
 0.34 
 
 0.36936 
 
 37037 
 
 37137 
 
 37238 
 
 37338 
 
 37438 
 
 37538 
 
 37638 
 
 37738 
 
 37838 
 
 0.35 
 
 0.37938 
 
 38038 
 
 38138 
 
 38237 
 
 38337 
 
 38436 
 
 38536 
 
 38635 
 
 38735 
 
 38834 
 
 0.36 
 
 0.38933 
 
 39032 
 
 39131 
 
 39230 
 
 39329 
 
 39428 
 
 39526 
 
 39625 
 
 39724 
 
 39822 
 
 0.37 
 
 0.39921 
 
 40019 
 
 40117 
 
 40215 
 
 40314 
 
 40412 
 
 40510 
 
 40608 
 
 40705 
 
 40803 
 
 0.38 
 
 0.40901 
 
 40999 
 
 41096 
 
 41194 
 
 41291 
 
 41388 
 
 41486 
 
 41583 
 
 41680 
 
 41777 
 
 0.39 
 
 0.41874 
 
 41971 
 
 42068 
 
 42164 
 
 42261 
 
 42358 
 
 42454 
 
 42550 
 
 42647 
 
 42743 
 
 0.40 
 
 0.42839 
 
 42935 
 
 43031 
 
 43127 
 
 43223 
 
 43319 
 
 43415 
 
 43510 
 
 43606 
 
 43701 
 
 0.41 
 
 0.43797 
 
 43892 
 
 43988 
 
 44083 
 
 44178 
 
 44273 
 
 44368 
 
 44463 
 
 44557 
 
 44652 
 
 0.42 
 
 0.44747 
 
 44841 
 
 44936 
 
 45030 
 
 45124 
 
 45219 
 
 45313 
 
 45407 
 
 45501 
 
 45595 
 
 0.43 
 
 0.45689 
 
 45782 
 
 45876 
 
 45970 
 
 46063 
 
 46157 
 
 46250 
 
 46343 
 
 46436 
 
 46529 
 
 0.44 
 
 0.46623 
 
 46715 
 
 46808 
 
 46901 
 
 46994 
 
 47086 
 
 47179 
 
 47271 
 
 47364 
 
 47456 
 
 0.45 
 
 0.47548 
 
 47640 
 
 47732 
 
 47824 
 
 47916 
 
 48008 
 
 48100 
 
 48191 
 
 48283 
 
 48374 
 
 0.46 
 
 0.48466 
 
 48557 
 
 48648 
 
 48739 
 
 48830 
 
 48921 
 
 49012 
 
 49103 
 
 49193 
 
 49284 
 
 0.47 
 
 0.49375 
 
 49465 
 
 49555 
 
 49646 
 
 49736 
 
 49826 
 
 49916 
 
 50006 
 
 50096 
 
 50185 
 
 0.48 
 
 0.50275 
 
 50365 
 
 50454 
 
 50543 
 
 50633 
 
 50722 
 
 50811 
 
 50900 
 
 50989 
 
 51078 
 
 0.49 
 
 0.51167 
 
 51256 
 
 51344 
 
 51433 
 
 51521 
 
 51609 
 
 51698 
 
 51786 
 
 51874 
 
 51962 
 
 1 
 
TABLES. 117 
 
 The Probability Integral. 
 
 123456789 
 
 0.52050 52138 52226 52313 52401 52488 52576 52663 52750 52837 
 
 0.52924 53011 53098 53185 53272 53358 53445 53531 53617 5370+ 
 
 0.53790 53876 53962 54048 54134 54219 54305 54390 54476 54561 
 
 0.54646 54732 54817 54902 54987 55071 55156 55241 55325 55410 
 
 0.55494 55578 55662 55746 55830 55914 55998 56082 56165 56249 
 
 0.56332 56416 56499 56582 56665 56748 56831 56914 56996 57079 
 
 0.57162 57244 57326 57409 57491 57573 57655 57737 57818 57900 
 
 0.57982 58063 58144 58226 58307 58388 58469 58550 58631 58712 
 
 0.58792 58873 58953 59034 59114 59194 59274 59354 59434 59514 
 
 0.59594 59673 59753 59832 59912 59991 60070 60149 60228 60307 
 
 0.60386 60464 60543 60621 60700 60778 60856 60934 61012 61090 
 
 0.61168 61246 61323 61401 61478 61556 61633 61710 61787 61864 
 
 0.61941 62018 62095 62171 62248 62324 62400 62477 62553 62629 
 
 0.62705 62780 62856 62932 63007 63083 63158 63233 63309 63384 
 
 0.63459 63533 63608 63683 63757 63832 63906 63981 64055 64129 
 
 0.64203 64277 64351 64424 64498 64572 64645 64718 64791 64865 
 
 0.64938 65011 65083 65156 65229 65301 65374 65446 65519 65591 
 
 0.65663 65735 65807 65878 65950 66022 66093 66165 66236 66307 
 
 0.66378 66449 66520 66591 66662 66732 66803 66873 66944 67014 
 
 0.67084 67154 67224 67294 67364 67433 67503 67572 67642 67711 
 
 0.67780 67849 67918 67987 68056 68125 68193 68262 68330 68398 
 
 0.68467 68535 68603 68671 68/38 68806 68874 68941 69009 69076 
 
 0.69143 69210 69278 69344 69411 69478 69545 69611 6%78 69744 
 
 0.69S10 69877 69943 70009 70075 70140 70206 70272 70337 70403 
 
 0.70468 70533 70598 70663 70728 70793 70858 70922 70987 71051 
 
 0.71116 71180 71244 71308 71372 71436 71500 71563 71627 71690 
 
 0.71754 71817 71880 71943 72006 72069 72132 72195 72257 72320 
 
 0.72382 72444 72507 72569 72631 72693 72755 72816 72878 72940 
 
 0.73001 73062 73124 73185 73246 73307 73368 73429 73489 73550 
 
 0.73610 73671 73731 73791 73851 73911 73971 74031 74091 74151 
 
 0.74210 74270 74329 74388 74447 74506 74565 74624 74683 74742 
 
 0.74800 74859 74917 74976 75034 75092 75150 75208 75266 75323 
 
 0.75381 75439 75496 75553 75611 75668 75725 75782 75839 75896 
 
 0.75952 76009 76066 76122 76178 76234 76291 76347 76403 76459 
 
 0.76514 76570 76626 76681 76736 76792 76847 76902 76957 77012 
 
 0.77067 77122 77176 77231 77285 77340 77394 77448 77502 77556 
 
 0.77610 77664 77718 77771 77825 77878 77932 77985 78038 78091 
 
 0.78144 78197 78250 78302 78355 78408 78460 78512 78565 78617 
 
 0.78669 78721 78773 78824 78876 78928 78979 79031 79082 79133 
 
 0.79184 79235 79286 79337 79388 79439 79489 79540 79590 79641 
 
 0.79691 79741 79791 79841 79891 79941 79990 80040 80090 80139 
 
 0.80188 80238 80287 80336 80385 80434 80482 80531 80580 80628 
 
 0.80677 80725 80773 80822 80870 80918 80966 81013 81061 81109 
 
 0.81156 81204 81251 81299 81346 81393 81440 81487 81534 81580 
 
 0.81627 81674 81720 81767 81813 81859 81905 81951 81997 82043 
 
 0.82089 82135 82180 82226 82271 82317 82362 82407 82452 82497 
 
 0.82542 82587 82632 82677 82721 82766 82810 82855 82899 82943 
 
 0.82987 83031 83075 83119 83162 83206 83250 83293 83337 83380 
 
 0.83423 83466 83509 83552 83595 83638 83681 83723 83766 83808 
 
 0.83851 83893 83935 83977 84020 84061 84103 84145 84187 84229 
 
118 
 
 TABLES. 
 
 The Probability Integral. 
 
 "^ dx. 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ^ 
 
 1 
 
 8 
 
 9 
 
 1.00 
 
 0.84270 
 
 84312 
 
 84353 
 
 84394 
 
 84435 
 
 84477 
 
 84518 
 
 84559 
 
 84600 
 
 84640 
 
 1.01 
 
 0.84681 
 
 84722 
 
 84762 
 
 84803 
 
 84843 
 
 84883 
 
 84924 
 
 84964 
 
 85004 
 
 85044 
 
 1.02 
 
 0.85084 
 
 85124 
 
 85163 
 
 85203 
 
 85243 
 
 85282 
 
 85322 
 
 85361 
 
 85400 
 
 85439 
 
 1.03 
 
 0.85478 
 
 85517 
 
 85556 
 
 85595 
 
 85634 
 
 85673 
 
 85711 
 
 85750 
 
 85788 
 
 85827 
 
 1.04 
 
 0.85865 
 
 85903 
 
 85941 
 
 85979 
 
 86017 
 
 86055 
 
 86093 
 
 86131 
 
 86169 
 
 86206 
 
 1.05 
 
 0.86244 
 
 86281 
 
 86318 
 
 86356 
 
 86393 
 
 86430 
 
 86467 
 
 86504 
 
 86541 
 
 86578 
 
 1.06 
 
 0.86614 
 
 86651 
 
 86688 
 
 86724 
 
 86760 
 
 86797 
 
 86833 
 
 86869 
 
 86905 
 
 86941 
 
 1.07 
 
 0.86977 
 
 87013 
 
 87049 
 
 87085 
 
 87120 
 
 87156 
 
 87191 
 
 87227 
 
 87262 
 
 87297 
 
 1.08 
 
 0.87333 
 
 87368 
 
 87403 
 
 87438 
 
 87473 
 
 87507 
 
 87542 
 
 87577 
 
 87611 
 
 87646 
 
 1.09 
 
 0.87680 
 
 87715 
 
 87749 
 
 87783 
 
 87817 
 
 87851 
 
 87885 
 
 87919 
 
 87953 
 
 87987 
 
 1.10 
 
 0.88021 
 
 88054 
 
 880SS 
 
 88121 
 
 88155 
 
 SS188 
 
 88221 
 
 88254 
 
 88287 
 
 88320 
 
 1.11 
 
 0.88353 
 
 88386 
 
 88419 
 
 88452 
 
 88484 
 
 88517 
 
 88549 
 
 88582 
 
 88614 
 
 88647 
 
 1.12 
 
 0.88679 
 
 88711 
 
 88743 
 
 88775 
 
 88807 
 
 8SS39 
 
 88871 
 
 88902 
 
 88934 
 
 88966 
 
 1.13 
 
 0.88997 
 
 89029 
 
 89060 
 
 89091 
 
 89122 
 
 89] 54 
 
 89185 
 
 89216 
 
 89247 
 
 89277 
 
 1.14 
 
 0.89308 
 
 89339 
 
 89370 
 
 89400 
 
 89431 
 
 89461 
 
 89492 
 
 89522 
 
 89552 
 
 89582 
 
 1.15 
 
 0.89612 
 
 89642 
 
 89672 
 
 89702 
 
 89732 
 
 89762 
 
 89792 
 
 89821 
 
 89851 
 
 89880 
 
 1.16 
 
 0.89910 
 
 89939 
 
 89968 
 
 89997 
 
 90027 
 
 90056 
 
 90085 
 
 90114 
 
 90142 
 
 90171 
 
 1.17 
 
 0.90200 
 
 90229 
 
 90257 
 
 90286 
 
 90314 
 
 90343 
 
 90371 
 
 90399 
 
 90428 
 
 90456 
 
 1.18 
 
 0.90484 
 
 90512 
 
 90540 
 
 90568 
 
 90595 
 
 90623 
 
 90651 
 
 90678 
 
 90706 
 
 90733 
 
 1.19 
 
 0.90761 
 
 90788 
 
 90815 
 
 90843 
 
 90870 
 
 90897 
 
 90924 
 
 90951 
 
 90978 
 
 91005 
 
 1.20 
 
 0.91031 
 
 91058 
 
 91085 
 
 91111 
 
 91138 
 
 91164 
 
 91191 
 
 91217 
 
 91243 
 
 91269 
 
 1.21 
 
 0.91296 
 
 91322 
 
 91348 
 
 91374 
 
 91399 
 
 91425 
 
 91451 
 
 91477 
 
 91502 
 
 91528 
 
 1.22 
 
 0.91553 
 
 91579 
 
 91604 
 
 91630 
 
 91655 
 
 91680 
 
 91705 
 
 91730 
 
 91755 
 
 91780 
 
 1.23 
 
 0.91805 
 
 91830 
 
 91855 
 
 91879 
 
 91904 
 
 91929 
 
 91953 
 
 91978 
 
 92002 
 
 92026 
 
 1.24 
 
 0.92051 
 
 92075 
 
 92099 
 
 92123 
 
 92147 
 
 92171 
 
 92195 
 
 92219 
 
 92243 
 
 92266 
 
 1.25 
 
 0.92290 
 
 92314 
 
 92337 
 
 92361 
 
 92384 
 
 92408 
 
 92431 
 
 92454 
 
 92477 
 
 92500 
 
 1.26 
 
 0.92524 
 
 92547 
 
 92570 
 
 92593 
 
 92615 
 
 92638 
 
 92661 
 
 92684 
 
 92706 
 
 92729 
 
 1.27 
 
 0.92751 
 
 92774 
 
 92796 
 
 92819 
 
 92841 
 
 92863 
 
 92885 
 
 92907 
 
 92929 
 
 92951 
 
 1.28 
 
 0.92973 
 
 92995 
 
 93017 
 
 93039 
 
 93061 
 
 93082 
 
 93104 
 
 93126 
 
 93147 
 
 93168 
 
 1.29 
 
 0.93190 
 
 93211 
 
 93232 
 
 93254 
 
 93275 
 
 93296 
 
 93317 
 
 93338 
 
 93359 
 
 93380 
 
 1.30 
 
 0.93401 
 
 93422 
 
 93442 
 
 93463 
 
 93484 
 
 93504 
 
 93525 
 
 93545 
 
 93566 
 
 93586 
 
 1.31 
 
 0.93606 
 
 93627 
 
 93647 
 
 93667 
 
 93687 
 
 93707 
 
 93727 
 
 93747 
 
 93767 
 
 93787 
 
 1.32 
 
 0.93807 
 
 93826 
 
 93846 
 
 93866 
 
 93885 
 
 93905 
 
 93924 
 
 93944 
 
 93963 
 
 93982 
 
 1.33 
 
 0.94002 
 
 94021 
 
 94040 
 
 94059 
 
 94078 
 
 94097 
 
 94116 
 
 94135 
 
 94154 
 
 94173 
 
 1.34 
 
 0.94191 
 
 94210 
 
 94229 
 
 94247 
 
 94266 
 
 94284 
 
 94303 
 
 94321 
 
 94340 
 
 94358 
 
 1.35 
 
 0.94376 
 
 94394 
 
 94413 
 
 94431 
 
 94449 
 
 94467 
 
 94485 
 
 94503 
 
 94521 
 
 94538 
 
 1.36 
 
 0.94556 
 
 94574 
 
 94592 
 
 94609 
 
 94627 
 
 94644 
 
 94662 
 
 94679 
 
 94697 
 
 94714 
 
 1.37 
 
 0.94731 
 
 94748 
 
 94766 
 
 94783 
 
 94800 
 
 94817 
 
 94834 
 
 94851 
 
 94868 
 
 94885 
 
 1.38 
 
 0.94902 
 
 94918 
 
 94935 
 
 94952 
 
 94968 
 
 94985 
 
 95002 
 
 95018 
 
 95035 
 
 95051 
 
 1.39 
 
 0.95067 
 
 95084 
 
 95100 
 
 95116 
 
 95132 
 
 95148 
 
 95165 
 
 95181 
 
 95197 
 
 95213 
 
 1.40 
 
 0.95229 
 
 95244 
 
 95260 
 
 95276 
 
 95292 
 
 95307 
 
 95323 
 
 95339 
 
 95354 
 
 95370 
 
 1.41 
 
 0.95385 
 
 95401 
 
 95416 
 
 95431 
 
 95447 
 
 95462 
 
 95477 
 
 95492 
 
 95507 
 
 95523 
 
 1.42 
 
 0.95538 
 
 95553 
 
 95568 
 
 95582 
 
 95597 
 
 95612 
 
 95627 
 
 95642 
 
 95656 
 
 95671 
 
 1.43 
 
 0.95686 
 
 95700 
 
 95715 
 
 95729 
 
 95744 
 
 95758 
 
 95773 
 
 95787 
 
 95801 
 
 95815 
 
 1.44 
 
 0.95830 
 
 95844 
 
 95858 
 
 95872 
 
 95886 
 
 95900 
 
 95914 
 
 95928 
 
 95942 
 
 95956 
 
 1.45 
 
 0.95970 
 
 95983 
 
 95997 
 
 96011 
 
 96024 
 
 96038 
 
 96051 
 
 96065 
 
 96078 
 
 96092 
 
 1.46 
 
 0.96105 
 
 96119 
 
 96132 
 
 96145 
 
 96159 
 
 96172 
 
 96185 
 
 96198 
 
 96211 
 
 96224 
 
 1.47 
 
 0.96237 
 
 96250 
 
 96263 
 
 96276 
 
 96289 
 
 96302 
 
 96315 
 
 96327 
 
 96340 
 
 96353 
 
 1.48 
 
 0.96365 
 
 96378 
 
 96391 
 
 96403 
 
 96416 
 
 96428 
 
 96440 
 
 96453 
 
 96465 
 
 96478 
 
 1.49 
 
 0.96490 
 
 96502 
 
 96514 
 
 96526 
 
 96539 
 
 96551 
 
 96563 
 
 96575 
 
 96587 
 
 96599 
 
TABLES. 
 
 119 
 
 The Probability Integral. 
 
 X 
 
 2 4 6 8 
 
 X 
 
 2 4 6 8 
 
 1.50 
 
 0.96611 96634 96658 96681 96705 
 
 2.00 
 
 0.99532 99536 99540 99544 99548 
 
 1.51 
 
 0.96728 96751 96774 96796 96819 
 
 2.01 
 
 0.99552 99556 99560 99564 99568 
 
 1.52 
 
 0.96841 96S64 96886 96908 96930 
 
 2.02 
 
 0.99572 99576 99580 99583 99587 
 
 1.53 
 
 0.96952 96973 96995 97016 97037 
 
 2.03 
 
 0.99591 99594 99598 99601 99605 
 
 1.54 
 
 0.97059 97080 97100 97121 97142 
 
 2.04 
 
 0.99609 99612 99616 99619 99622 
 
 1.55 
 
 0.97162 97183 97203 97223 97243 
 
 2.05 
 
 0.99626 99629 99633 99636 99639 
 
 1.56 
 
 0.97263 97283 97302 97322 97341 
 
 2.06 
 
 0.99642 99646 99649 99652 99655 
 
 1.57 
 
 0.97360 97379 9739S 97417 97436 
 
 2.07 
 
 0.99658 99661 99664 99667 99670 
 
 1.58 
 
 0.97455 97473 97492 97510 97528 
 
 2. OS 
 
 0.99673 99676 99679 996S2 99685 
 
 1.59 
 
 0.97546 97564 97582 97600 97617 
 
 2.09 
 
 0.99688 99691 99694 99697 99699 
 
 1.60 
 
 0.97635 97652 97670 97687 97704 
 
 2.10 
 
 0.99702 99705 99707 99710 99713 
 
 1.61 
 
 0.97721 97738 97754 97771 97787 
 
 2.11 
 
 0.99715 99718 99721 99723 99726 
 
 1.62 
 
 0.97804 97820 97836 97852 97868 
 
 2.12 
 
 0.99728 99731 99733 99736 99738 ' 
 
 1.63 
 
 0.978S4 97900 97916 97931 97947 
 
 2.13 
 
 0.99741 99743 99745 99748 99750 
 
 1.64 
 
 0.97962 97977 97993 98008 98023 
 
 2.14 
 
 0.99753 99755 99757 99759 99762 
 
 1.65 
 
 0.9S03S 98052 98067 98082 98096 
 
 2.15 
 
 0.99764 99766 99768 99770 99773 
 
 1.66 
 
 0.98110 98125 98139 98153 98167 
 
 2.16 
 
 0.99775 99777 99779 997S1 99783 
 
 1.67 
 
 0.98181 98195 98209 98222 98236 
 
 2.17 
 
 0.99785 99787 99789 99791 99793 
 
 1.68 
 
 0.98249 9S263 9S276 98289 98302 
 
 2.18 
 
 0.99795 99797 99799 99801 99803 
 
 1.69 
 
 0.9S315 98328 9S341 98354 98366 
 
 2.19 
 
 0.99805 99806 99808 99810 99812 
 
 1.70 
 
 0.98379 98392 98404 98416 98429 
 
 2.20 
 
 0.99814 99815 99817 99819 99821 
 
 1.71 
 
 0.98441 9S453 98465 98477 98489 
 
 2.21 
 
 0.99822 99824 99826 99827 99829 
 
 1.72 
 
 0.98500 98512 98524 98535 98546 
 
 2.22 
 
 0.99831 99832 99834 99836 99S37 
 
 1.73 
 
 0.98558 98569 98580 98591 98602 
 
 2.23 
 
 0.99839 99840 99842 99843 99845 
 
 1.74 
 
 0.98613 98624 98635 98646 98657 
 
 2.24 
 
 0.99846 99848 99849 99851 99852 
 
 1.75 
 
 0.98667 98678 98688 98699 98709 
 
 2.25 
 
 0.99854 99855 9985 7 99858 99859 
 
 1.76 
 
 0.98719 9S729 98739 98749 98759 
 
 2.26 
 
 0.99861 99862 99863 99865 99866 
 
 1.77 
 
 0.98769 98779 98789 98798 98808 
 
 2.27 
 
 0.99867 99869 99870 99871 99873 
 
 1.78 
 
 0.98817 98827 9SS36 98846 98855 
 
 2.28 
 
 0.99874 99875 99876 99877 99879 
 
 1.79 
 
 0.98864 98873 98882 98891 98900 
 
 2.29 
 
 0.99880 99881 99882 99883 99885 
 
 1.80 
 
 0.98909 98918 98927 98935 98944 
 
 2.30 
 
 0.99886 99887 99888 99889 99890 
 
 1.81 
 
 0.98952 98961 98969 98978 98986 
 
 2.31 
 
 0.99891 99892 99893 99894 99896 
 
 1.82 
 
 0.98994 99003 99011 99019 99027 
 
 2.32 
 
 0.99897 99898 99899 99900 99901 
 
 1.83 
 
 0.99035 99043 99050 99058 99066 
 
 2.33 
 
 0.99902 99903 99904 99905 99906 
 
 1.84 
 
 0.99074 99081 99089 99096 99104 
 
 2.34 
 
 0.99906 99907 99908 99909 99910 
 
 1.85 
 
 0.99111 99118 99126 99133 99140 
 
 2.35 
 
 0.99911 99912 99913 99914 99915 
 
 1.86 
 
 0.99147 99154 99161 99168 99175 
 
 2.36 
 
 0.99915 99916 99917 99918 99919 
 
 1.87 
 
 0.99182 99189 99196 99202 99209 
 
 2.37 
 
 0.99920 99920 99921 99922 99923 
 
 l.SS 
 
 0.99216 99222 99229 99235 99242 
 
 2.38 
 
 0.99924 99924 99925 99926 99927 
 
 1.89 
 
 0.99248 99254 99261 99267 99273 
 
 2.39 
 
 0.99928 99928 99929 99930 99930 
 
 1.90 
 
 0.99279 99285 99291 99297 99303 
 
 2.40 
 
 0.99931 99932 99933 99933 99934 
 
 1.91 
 
 0.99309 99315 99321 99326 99332 
 
 2.41 
 
 0.99935 99935 99936 99937 99937 
 
 1.92 
 
 0.99338 99343 99349 99355 99360 
 
 2.42 
 
 0.99938 99939 99939 99940 99940 
 
 1.93 
 
 0.99366 99371 99376 99382 99387 
 
 2.43 
 
 0.99941 99942 99942 99943 99943 
 
 1.94 
 
 0.99392 99397 99403 99408 99413 
 
 2.44 
 
 0.99944 99945 99945 99946 99946 
 
 1.95 
 
 0.99418 99423 99428 99433 99438 
 
 2.45 
 
 0.99947 99947 99948 99949 99949 
 
 1.96 
 
 0.99443 99447 99452 99457 99462 
 
 2.46 
 
 0.99950 99950 99951 99951 99952 
 
 1.97 
 
 0.99466 99471 99476 99480 99485 
 
 2.47 
 
 0.99952 99953 99953 99954 99954 
 
 1.98 
 
 0.99489 99494 99498 99502 99507 
 
 2.48 
 
 0.99955 99955 99956 99956 99957 
 
 1.99 
 
 0.99511 99515 99520 99524 99528 
 
 2.49 
 
 0.99957 99958 99958 99958 99959 
 
 2.00 
 
 0.99532 99536 99540 99544 99548 
 
 2.50 
 
 0.99959 99960 99960 99961 99961 
 
120 
 
 TABLES. 
 
 The Probability Integral. 
 
 2 
 
 -x2 
 
 
 ) 
 
 X 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2.5 
 
 0.99959 
 
 99961 
 
 99963 
 
 99965 
 
 99967 
 
 99969 
 
 99971 
 
 99972 
 
 99974 
 
 99975 
 
 2.6 
 
 0.99976 
 
 99978 
 
 99979 
 
 99980 
 
 99981 
 
 99982 
 
 99983 
 
 99984 
 
 99985 
 
 99986 
 
 2.7 
 
 0.99987 
 
 99987 
 
 99988 
 
 99989 
 
 99989 
 
 99990 
 
 99991 
 
 99991 
 
 99992 
 
 99992 
 
 2.8 
 
 0.99992 
 
 99993 
 
 99993 
 
 99994 
 
 99994 
 
 99994 
 
 99995 
 
 99995 
 
 99995 
 
 99996 
 
 2.9 
 
 0.99996 
 
 99996 
 
 99996 
 
 99997 
 
 99997 
 
 99997 
 
 99997 
 
 99997 
 
 99997 
 
 99998 
 
 3.0 
 
 0.99998 
 
 99998 
 
 99998 
 
 99998 
 
 99998 
 
 99998 
 
 99998 
 
 99998 
 
 99999 
 
 99999 
 
 The value, /, of the Probability Integral may always be found from the convergent series 
 
 , 1 / afi a;5 a-' 
 /= — = [x — , 
 
 ■v^V 3-l!^5-2! 7-3! 
 
 ■)■ 
 
 but for large values of x, the semiconvergent series 
 
 x-^-a\ 2x2 (2x2)3 (2x2)3^ ) 
 
 is convenient. 
 
 i 
 
TABLES. 
 
 121 
 
 Values of the Complete Elliptic Integrals, JTand E, for Different 
 Values of the Modulus, k. 
 
 K= p. ^^ ; E= pVl-A;2sin-z. 
 
 »/o Vl — kr sin'' z *yo 
 
 dz. 
 
 sin-ifc 
 
 E 
 
 E 
 
 sin-iA; 
 
 K 
 
 E 
 
 1 
 sin-ifc 
 
 K 
 
 E 
 
 0° 
 
 1.5708 
 
 1.5708 
 
 30° 
 
 1.6858 
 
 1.4675 
 
 60° 
 
 2.1565 
 
 1.2111 
 
 1° 
 
 1.5709 
 
 1.5707 
 
 31° 
 
 1.6941 
 
 1.4608 
 
 61° 
 
 2.1842 
 
 1.2015 
 
 2° 
 
 1.5713 
 
 1.5703 
 
 32° 
 
 1.7028 
 
 1.4539 
 
 62° 
 
 2.2132 
 
 1.1920 
 
 3° 
 
 1.5719 
 
 1.5697 
 
 33° 
 
 1.7119 
 
 1.4469 
 
 63° 
 
 2.2435 
 
 1.1826 
 
 4° 
 
 1.5727 
 
 1.5689 
 
 34° 
 
 1.7214 
 
 1.4397 
 
 64° 
 
 2.2754 
 
 1.1732 
 
 5° 
 
 1.5738 
 
 1.5678 
 
 35° 
 
 1.7312 
 
 1,4323 
 
 65° 
 
 2.308S 
 
 1.1638 
 
 6° 
 
 1.5751 
 
 1.5665 
 
 36° 
 
 1.7415 
 
 1.4248 
 
 66° 
 
 2.3439 
 
 1.1545 
 
 7° 
 
 1.5767 
 
 1.5649 
 
 37° 
 
 1.7522 
 
 1.4171 
 
 67° 
 
 2.3809 
 
 1.1453 
 
 8° 
 
 1.5785 
 
 1.5632 
 
 38° 
 
 1.7633 
 
 1.4092 
 
 68° 
 
 2.4198 
 
 1.1362 
 
 9° 
 
 1.5S05 
 
 1.5611 
 
 39° 
 
 1.7748 
 
 1.4013 
 
 69° 
 
 2.4610 
 
 1.1272 
 
 10° 
 
 1.5828 
 
 1.5589 
 
 40° 
 
 1.7868 
 
 1.3931 
 
 70° 
 
 2.5046 
 
 1.1184 
 
 11° 
 
 1.5854 
 
 1.5564 
 
 41° 
 
 1.7992 
 
 1.3849 
 
 71° 
 
 2.5507 
 
 1.1096 
 
 12° 
 
 1.5882 
 
 1.5537 
 
 42° 
 
 1.8122 
 
 1.3765 
 
 72° 
 
 2.5998 
 
 1.1011 
 
 13° 
 
 1.5913 
 
 1.5507 
 
 43° 
 
 1.8256 
 
 1.3680 
 
 73° 
 
 2.6521 
 
 1.0927 
 
 14° 
 
 1.5946 
 
 1.5476 
 
 44° 
 
 1.8396 
 
 1.3594 
 
 74° 
 
 2.7081 
 
 1.0S44 
 
 15° 
 
 1.5981 
 
 1.5442 
 
 45° 
 
 1.S541 
 
 1.3506 
 
 75° 
 
 2.7681 
 
 1.0764 
 
 16° 
 
 1.6020 
 
 1.5405 
 
 46° 
 
 1.S691 
 
 1.3418 
 
 76° 
 
 2.8327 
 
 1.0686 
 
 17° 
 
 1.6061 
 
 1.5367 
 
 47° 
 
 1.SS48 
 
 1.3329 
 
 77° 
 
 2.9026 
 
 1.0611 
 
 18° 
 
 1.6105 
 
 1.5326 
 
 48° 
 
 1.9011 
 
 1.3238 
 
 78° 
 
 2.9786 
 
 1.0538 
 
 19° 
 
 1.6151 
 
 1.5283 
 
 49° 
 
 1.9180 
 
 1.3147 
 
 79° 
 
 3.0617 
 
 1.0468 
 
 20° 
 
 1.6200 
 
 1.5238 
 
 50° 
 
 1.9356 
 
 1.3055 
 
 80° 
 
 3.1534 
 
 1.0401 
 
 21° 
 
 1.6252 
 
 1.5191 
 
 51° 
 
 1.9539 
 
 1.2963 
 
 81° 
 
 3.2553 
 
 1.0338 
 
 22° 
 
 1.6307 
 
 1.5141 
 
 52° 
 
 1.9729 
 
 1.2S70 
 
 82° 
 
 3.3699 
 
 1.0278 
 
 23° 
 
 1.6365 
 
 1.5090 
 
 53° 
 
 1.9927 
 
 1.2776 
 
 83° 
 
 3.5004 
 
 1.0223 
 
 24° 
 
 1.6426 
 
 1.5037 
 
 54° 
 
 2.0133 
 
 1.26S1 
 
 84° 
 
 3.6519 
 
 1.0172 
 
 25° 
 
 1.6490 
 
 1.4981 
 
 55° 
 
 2.0347 
 
 1.2587 
 
 85° 
 
 3.8317 
 
 1.0127 
 
 26° 
 
 1.6557 
 
 1.4924 
 
 56° 
 
 2.0571 
 
 1.2492 
 
 86° 
 
 4.0528 
 
 1.0086 
 
 27° 
 
 1.6627 
 
 1.4864 
 
 57° 
 
 2.080+ 
 
 1.2397 
 
 87° 
 
 4.3387 
 
 1.0053 
 
 28° 
 
 1.6701 
 
 1.4803 
 
 58° 
 
 2.101-7 
 
 1.2301 
 
 88° 
 
 4.7427 
 
 1.0026 
 
 29° 
 
 1.6777 
 
 1.4740 
 
 59° 
 
 2.1300 
 
 1.2206 
 
 89° 
 
 5.4349 
 
 1.0008 
 
122 
 
 TABLES. 
 
 Values of F{k, <t>) for Certain Values of k and ^t, 
 
 dz 
 
 ^<*'*>=XV 
 
 A;2 sin2 z 
 
 <t> 
 
 a = sin-^k. 
 
 0° 
 
 1(P 
 
 15° 
 
 30° 
 
 45° 
 
 60° 
 
 75° 
 
 80° 
 
 90° 
 
 1° 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 2° 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 3° 
 
 0.0524 
 
 0.0524 
 
 0.0524 
 
 0.0524 
 
 0.0524 
 
 0.0524 
 
 0.0524 
 
 0.0524 
 
 0.0524 
 
 40 
 
 0.0698 
 
 0.0698 
 
 0.069S 
 
 0.0698 
 
 0.0698 
 
 0.0699 
 
 0.0699 
 
 0.0699 
 
 0.0699 
 
 5° 
 
 0.0873 
 
 0.0873 
 
 0.0873 
 
 0.0873 
 
 0.0873 
 
 0.0S74 
 
 0.0874 
 
 0.0874 
 
 0.0874 
 
 10° 
 
 0.1745 
 
 0.1746 
 
 0.1746 
 
 0.1748 
 
 0.1750 
 
 0.1752 
 
 0.1754 
 
 0.1754 
 
 0.1754 
 
 15° 
 
 0.2618 
 
 0.2619 
 
 0.2620 
 
 0.2625 
 
 0.2633 
 
 0.2641 
 
 0.2646 
 
 0.2647 
 
 0.2648 
 
 20° 
 
 0.3491 
 
 0.3493 
 
 0.3495 
 
 0.3508 
 
 0.3526 
 
 0.3545 
 
 0.3559 
 
 0.3562 
 
 0.3564 
 
 25° 
 
 0.4363 
 
 0.4367 
 
 0.4372 
 
 0.4397 
 
 0.4433 
 
 0.4470 
 
 0.4498 
 
 0.4504 
 
 0.4509 
 
 30° 
 
 0.5236 
 
 0.5243 
 
 0.5251 
 
 0.5294 
 
 0.5356 
 
 0.5422 
 
 0.5474 
 
 0.5484 
 
 0.5493 
 
 35° 
 
 0.6109 
 
 0.6119 
 
 0.6132 
 
 0.6200 
 
 0.6300 
 
 0.6408 
 
 0.6495 
 
 0.6513 
 
 0.6528 
 
 40° 
 
 0.6981 
 
 0.6997 
 
 0.7016 
 
 0.7116 
 
 0.7267 
 
 0.7436 
 
 0.7574 
 
 0.7604 
 
 0.7629 
 
 45° 
 
 0.7854 
 
 0.7876 
 
 0.7902 
 
 0.8044 
 
 0.8260 
 
 0.8512 
 
 0.8727 
 
 0.8774 
 
 0.8814 
 
 50° 
 
 0.8727 
 
 0.8756 
 
 0.8792 
 
 0.8982 
 
 0.9283 
 
 0.9646 
 
 0.9971 
 
 1.0044 
 
 1.0107 
 
 55° 
 
 0.9599 
 
 0.9637 
 
 0.9683 
 
 0.9933 
 
 1.0337 
 
 1.0848 
 
 1.1331 
 
 1.1444 
 
 1.1542 
 
 60° 
 
 l.(H72 
 
 1.0519 
 
 1.0577 
 
 1.0896 
 
 1.1424 
 
 1.2125 
 
 1.2837 
 
 1.3014 
 
 1.3170 
 
 65° 
 
 1.1345 
 
 1.1402 
 
 1.1474 
 
 1.1869 
 
 1.2545 
 
 1.3489 
 
 1.4532 
 
 1.4810 
 
 1.5064 
 
 70° 
 
 1.2217 
 
 1.2286 
 
 1.2373 
 
 1.2853 
 
 1.3697 
 
 1.4944 
 
 1.6468 
 
 1.6918 
 
 1.7354 
 
 75° 
 
 1.3090 
 
 1.3171 
 
 1.3273 
 
 1.3846 
 
 1.4S79 
 
 1.6492 
 
 1.8714 
 
 1.9468 
 
 2.0276 
 
 80° 
 
 1.3963 
 
 1.4056 
 
 1.4175 
 
 1.4846 
 
 1.6085 
 
 1.8125 
 
 2.1339 
 
 2.2653 
 
 2.4362 
 
 85° 
 
 1.4835 
 
 1.4942 
 
 1.5078 
 
 1.5850 
 
 1.7308 
 
 1.9826 
 
 2.4366 
 
 2.6694 
 
 3.1313 
 
 86° 
 
 1.5010 
 
 1.5120 
 
 1.5259 
 
 1.6052 
 
 1.7554 
 
 2.0172 
 
 2.5013 
 
 2.7612 
 
 3.3547 
 
 87° 
 
 1.5184 
 
 1.5297 
 
 1.5439 
 
 1.6253 
 
 1.7801 
 
 2.0519 
 
 2.5670 
 
 2.8561 
 
 3.6425 
 
 88° 
 
 1.5359 
 
 1.5474 
 
 1.5620 
 
 1.6454 
 
 1.8047 
 
 2.0867 
 
 2.6336 
 
 2.9537 
 
 4.0481 
 
 89° 
 
 1.5533 
 
 1.5651 
 
 1.5S01 
 
 1.6656 
 
 1.8294 
 
 2.1216 
 
 2.7007 
 
 3.0530 
 
 4.7414 
 
 90° 
 
 1.5708 
 
 1.5828 
 
 1.5981 
 
 1.6858 
 
 1.8541 
 
 2.1565 
 
 2.7681 
 
 3.1534 
 
 Inf. 
 
TABLES. 
 
 123 
 
 Values of E(k, 4>) for Certain Values of k and 0. 
 E{k, ^) = I Vl - fc2 sin2 z • dz. 
 
 : 
 
 a = sin-i&. 
 
 0° 
 
 10° 
 
 15° 
 
 30° 
 
 45° 
 
 60° 
 
 75° 
 
 80° 
 
 90° 
 
 P 
 
 0.0174 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 0.0174 
 
 2» 
 
 0.0349 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 0.0349 
 
 3» 
 
 0.0524 0.0524 
 
 0.0524 
 
 0.0524 
 
 0.0524 
 
 0.0523 
 
 0.0523 
 
 0.0523 
 
 0.0523 
 
 40 
 
 0.0698 
 
 0.0698 
 
 0.0698 
 
 0.0698 
 
 0.0698 
 
 0.0698 
 
 0.0698 
 
 0.0698 
 
 0.0698 
 
 5° 
 
 0.0873 
 
 0.0873 
 
 0.0873 
 
 0.0872 
 
 0.0872 
 
 0.0872 
 
 0.0S72 
 
 0.0872 
 
 0.0872 
 
 10° 
 
 0.1745 
 
 0.1745 
 
 0.1745 
 
 0.1743 
 
 0.1741 
 
 0.1739 
 
 0.1737 
 
 0.1737 
 
 0.1736 
 
 15° 
 
 0.2618 
 
 0.2617 
 
 0.2616 
 
 0.2611 
 
 0.2603 
 
 0.2596 
 
 0.2590 
 
 0.2589 
 
 0.2588 
 
 20° 
 
 0.3491 
 
 0.3489 
 
 0.3486 
 
 0.3473 
 
 0.3456 
 
 0.3438 
 
 0.3425 
 
 0.3422 
 
 0.3420 
 
 25° 
 
 0.4363 
 
 0.4359 
 
 0.4354 
 
 0.4330 
 
 0.4296 
 
 0.4261 
 
 0.4236 
 
 0.4230 
 
 0.4226 
 
 30° 
 
 0.5236 
 
 0.5229 
 
 0.5221 
 
 0.5179 
 
 0.5120 
 
 0.5061 
 
 0.5016 
 
 0.5007 
 
 0.5000 
 
 35° 
 
 0.6109 
 
 0.6098 
 
 0.6085 
 
 0.6019 
 
 0.5928 
 
 0.5833 
 
 0.5762 
 
 0.5748 
 
 0.5736 
 
 40° 
 
 0.6981 
 
 0.6966 
 
 0.6947 
 
 0.6851 
 
 0.6715 
 
 0.6575 
 
 0.6468 
 
 0.6446 
 
 0.6428 
 
 45° 
 
 0.7854 
 
 0.7832 
 
 0.7806 
 
 0.7672 
 
 0.7482 
 
 0.7282 
 
 0.7129 
 
 0.7097 
 
 0.7071 
 
 50° 
 
 0.8727 
 
 0.8698 
 
 0.8663 
 
 0.8483 
 
 0.8226 
 
 0.7954 
 
 0.7741 
 
 0.7697 
 
 0.7660 
 
 55° 
 
 0.9599 
 
 0.9562 
 
 0.9517 
 
 0.9284 
 
 0.8949 
 
 0.8588 
 
 0.8302 
 
 0.8242 
 
 0.8192 
 
 60° 
 
 1.0472 
 
 1.0426 
 
 1.0368 
 
 1.0076 
 
 0.9650 
 
 0.9184 
 
 0.8808 
 
 0.8728 
 
 0.8660 
 
 65° 
 
 1.1345 
 
 1.12SS 
 
 1.1218 
 
 1.0858 
 
 1.0329 
 
 0.9743 
 
 0.9258 
 
 0.9152 
 
 0.9063 
 
 70° 
 
 1.2217 
 
 1.2149 
 
 1.2065 
 
 1.1632 
 
 1.0990 
 
 1.0266 
 
 0.9652 
 
 0.9514 
 
 0.9397 
 
 75° 
 
 1.3090 
 
 1.3010 
 
 1.2911 
 
 1.2399 
 
 1.1635 
 
 1.0759 
 
 0.9992 
 
 0.9814 
 
 0.9659 
 
 80° 
 
 1.3963 
 
 1.3S70 
 
 1.3755 
 
 1.3161 
 
 1.2266 
 
 1.1225 
 
 1.0282 
 
 1.0054 
 
 0.9848 
 
 85° 
 
 1.4835 
 
 1.4729 
 
 1.4598 
 
 1.3919 
 
 1.2889 
 
 1.1673 
 
 1.0534 
 
 1.0244 
 
 0.9962 
 
 86° 
 
 1.5010 
 
 1.4901 
 
 1.4767 
 
 1.4070 
 
 1.3012 
 
 1.1761 
 
 1.0581 
 
 1.0277 
 
 0.9976 
 
 87° 
 
 1.5184 
 
 1.5073 
 
 1.4936 
 
 1.4221 
 
 1.3136 
 
 1.1848 
 
 1.0628 
 
 1.0309 
 
 0.9986 
 
 88° 
 
 1.5359 
 
 1.5245 
 
 1.5104 
 
 1.4372 
 
 1.3260 
 
 1.1936 
 
 1.0674 
 
 1.0340 
 
 0.9994 
 
 89° 
 
 1.5533 
 
 1.5417 
 
 1.5273 
 
 1.4524 
 
 1.3383 
 
 1.2023 
 
 1.0719 
 
 1.0371 
 
 0.9998 
 
 90« 
 
 1.5708 
 
 1.5589 
 
 1.5442 
 
 1.4675 
 
 1.3506 
 
 1.2111 
 
 1.0764 
 
 1.0401 
 
 1 
 
 1.0000 
 
124 
 
 TABLES. 
 
 Hyperbolic Functions. 
 
 1, 
 
 e^ 
 
 e-^ 
 
 sinhx 
 
 coshx 
 
 gdx 
 
 0.00 
 
 1.0000 
 
 1.0000 
 
 0.0000 
 
 1.0000 
 
 o!oooo 
 
 .01 
 
 1.0100 
 
 0.9900 
 
 .0100 
 
 1.0000 
 
 0.5729 
 
 .02 
 
 1.0202 
 
 .9802 
 
 .0200 
 
 1.0002 
 
 1.1458 
 
 .03 
 
 1.0305 
 
 .9704 
 
 .0300 
 
 1.0004 
 
 1.7186 
 
 .04 
 
 1.0408 
 
 .9608 
 
 .0400 
 
 1.0008 
 
 2.2912 
 
 .05 
 
 1.0513 
 
 .9512 
 
 .0500 
 
 1.0013 
 
 2.8636 
 
 .06 
 
 1.0618 
 
 .9418 
 
 .0600 
 
 1.0018 
 
 3.4357 
 
 .07 
 
 1.0725 
 
 .9324 
 
 .0701 
 
 1.0025 
 
 4.0074 
 
 .08 
 
 1.0833 
 
 .9231 
 
 .0801 
 
 1.0032 
 
 4.5788 
 
 .09 
 
 1.0942 
 
 .9139 
 
 .0901 
 
 1.0041 
 
 5.1497 
 
 .10 
 
 1.1052 
 
 .9048 
 
 .1002 
 
 1.0050 
 
 5.720 
 
 .11 
 
 1.1163 
 
 .8958 
 
 .1102 
 
 1.0061 
 
 6.290 
 
 .12 
 
 1.1275 
 
 .8869 
 
 .1203 
 
 1.0072 
 
 6.859 
 
 .13 
 
 1.1388 
 
 .8781 
 
 .1304 
 
 1.0085 
 
 7.428 
 
 .14 
 
 1.1503 
 
 .8694 
 
 .1405 
 
 1.0098 
 
 7.995 
 
 .15 
 
 1.1618 
 
 .8607 
 
 .1506 
 
 1.0113 
 
 8.562 
 
 .16 
 
 1.1735 
 
 .8521 
 
 .1607 
 
 1.0128 
 
 9.128 
 
 .17 
 
 1.1853 
 
 .8437 
 
 .1708 
 
 1.0145 
 
 9.694 
 
 .18 
 
 1.1972 
 
 .8353 
 
 .1810 
 
 1.0162 
 
 10.258 
 
 .19 
 
 1.2092 
 
 .8270 
 
 .1911 
 
 1.0181 
 
 10.821 
 
 .20 
 
 1.2214 
 
 .8187 
 
 .2013 
 
 1.0201 
 
 11.384 
 
 .21 
 
 1.2337 
 
 .8106 
 
 .2115 
 
 1.0221 
 
 11.945 
 
 .22 
 
 1.2461 
 
 .8025 
 
 .2218 
 
 1.0243 
 
 12.505 
 
 .23 
 
 1.2586 
 
 .7945 
 
 .2320 
 
 1.0266 
 
 13.063 
 
 .24 
 
 1.2712 
 
 .7866 
 
 .2423 
 
 1.0289 
 
 13.621 
 
 .25 
 
 1.2840 
 
 .7788 
 
 .2526 
 
 1.0314 
 
 14.177 
 
 .26 
 
 1.2969 
 
 .7711 
 
 .2629 
 
 1.0340 
 
 14.732 
 
 .27 
 
 1.3100 
 
 .7634 
 
 .2733 
 
 1.0367 
 
 15.285 
 
 .28 
 
 1.3231 
 
 .7558 
 
 .2837 
 
 1.0395 
 
 15.837 
 
 .29 
 
 1.3364 
 
 .7483 
 
 .2941 
 
 1.0423 
 
 16.388 
 
 .30 
 
 1.3499 
 
 .7408 
 
 .3045 
 
 1.0453 
 
 16.937 
 
 .31 
 
 1.3634 
 
 .7334 
 
 .3150 
 
 1.0484 
 
 17.484 
 
 .32 
 
 1.3771 
 
 .7261 
 
 .3255 
 
 1.0516 
 
 18.030 
 
 .33 
 
 1.3910 
 
 .7189 
 
 .3360 
 
 1.0549 
 
 18.573 
 
 .34 
 
 1.4049 
 
 .7118 
 
 .3466 
 
 1.0584 
 
 19.116 
 
 .35 
 
 1.4191 
 
 .7047 
 
 .3572 
 
 1.0619 
 
 19.656 
 
 .36 
 
 1.4333 
 
 .6977 
 
 .3678 
 
 1.0655 
 
 20.195 
 
 .37 
 
 1.4477 
 
 .6907 
 
 .3785 
 
 1.0692 
 
 20.732 
 
 .38 
 
 1.4623 
 
 .6839 
 
 .3892 
 
 1.0731 
 
 21.267 
 
 .39 
 
 1.4770 
 
 .6771 
 
 .4000 
 
 1.0770 
 
 21.800 
 
 .40 
 
 1.4918 
 
 .6703 
 
 .4108 
 
 1.0811 
 
 22.331 
 
 .41 
 
 1.5068 
 
 .6636 
 
 .4216 
 
 1.0852 
 
 22.859 
 
 .42 
 
 1.5220 
 
 .6570 
 
 .4325 
 
 1.0895 
 
 23.386 
 
 .43 
 
 1.5373 
 
 .6505 
 
 .4434 
 
 1.0939 
 
 23.911 
 
 .44 
 
 1.5527 
 
 .6440 
 
 .4543 
 
 1.0984 
 
 24.434 
 
 .45 
 
 1.5683 
 
 .6376 
 
 .4653 
 
 1.1030 
 
 24.955 
 
 .46 
 
 1.5841 
 
 .6313 
 
 .4764 
 
 1.1077 
 
 25.473 
 
 .47 
 
 1.6000 
 
 .6250 
 
 .4875 
 
 1.1125 
 
 25.989 
 
 .48 
 
 1.6161 
 
 .6188 
 
 .4986 
 
 1.1174 
 
 26.503 
 
 .49 
 
 1.6323 
 
 .6126 
 
 .5098 
 
 1.1225 
 
 27.015 
 
 0.50 
 
 1.648f 
 
 0.6065 
 
 0.5211 
 
 1.1276 
 
 27?524 
 
 Note. —This table is talien from Prof. Byerly's Treatise on Fourier's Series, published by Messrs. 
 Oiim& Co. 
 
TABLES. 
 
 125 
 
 Hyperbolic Functions. 
 
 X 
 
 e-^ 
 
 er-x 
 
 sinhx 
 
 coshx 
 
 gdx 
 
 0.50 
 
 1.6487 
 
 0.6065 
 
 0.5211 
 
 1.1276 
 
 27!524 
 
 .51 
 
 1.6653 
 
 .6005 
 
 .5324 
 
 1.1329 
 
 28.031 
 
 .52 
 
 1.6820 
 
 .5945 
 
 .5438 
 
 1.1383 
 
 28.535 
 
 .53 
 
 1.6989 
 
 .5886 
 
 .5552 
 
 1.1438 
 
 29.037 
 
 .54 
 
 1.7160 
 
 .5827 
 
 .5666 
 
 1.1494 
 
 29.537 
 
 .55 
 
 1.7333 
 
 .5770 
 
 .5782 
 
 1.1551 
 
 30.034 
 
 .56 
 
 1.7507 
 
 .5712 
 
 .5897 
 
 1.1609 
 
 30.529 
 
 .57 
 
 1.7683 
 
 .5655 
 
 .6014 
 
 1.1669 
 
 31.021 
 
 .58 
 
 1.7860 
 
 .5599 
 
 .6131 
 
 1.1730 
 
 31.511 
 
 .59 
 
 1.8040 
 
 .5543 
 
 .6248 
 
 1.1792 
 
 31.998 
 
 .60 
 
 1.8221 
 
 .5488 
 
 6367 
 
 1.1855 
 
 32.483 
 
 .61 
 
 1.8404 
 
 .5433 
 
 .6485 
 
 1.1919 
 
 32.965 
 
 .62 
 
 1.8589 
 
 .5379 
 
 .6605 
 
 1.1984 
 
 33.444 
 
 .63 
 
 1.S776 
 
 .5326 
 
 .6725 
 
 1.2051 
 
 33.921 
 
 .64 
 
 1.8965 
 
 .5273 
 
 .6846 
 
 1.2119 
 
 34.395 
 
 .65 
 
 1.9155 
 
 .5220 
 
 .6967 
 
 1.2188 
 
 34.867 
 
 .66 
 
 1.9348 
 
 .5169 
 
 .7090 
 
 1.2258 
 
 35.336 
 
 .67 
 
 1.9542 
 
 .5117 
 
 .7213 
 
 1.2330 
 
 35.802 
 
 -68 
 
 1.9739 
 
 .5066 
 
 .7336 
 
 1.2402 
 
 36.265 
 
 -69 
 
 1.9937 
 
 .5016 
 
 .7461 
 
 1.2476 
 
 36.726 
 
 .70 
 
 2.0138 
 
 .4966 
 
 .7586 
 
 1.2552 
 
 37.183 
 
 -71 
 
 2.0340 
 
 .4916 
 
 .7712 
 
 1.2628 
 
 37.638 
 
 -72 
 
 2.0544 
 
 .4867 
 
 .7838 
 
 1.2706 
 
 38.091 
 
 -73 
 
 2.0751 
 
 .4819 
 
 .7966 
 
 1.2785 
 
 38.540 
 
 .74 
 
 2.0959 
 
 .4771 
 
 .8094 
 
 1.2865 
 
 38.987 
 
 .75 
 
 2.1170 
 
 .4724 
 
 .8223 
 
 1.2947 
 
 39.431 
 
 .76 
 
 2.1383 
 
 .4677 
 
 .8353 
 
 1.3030 
 
 39.872 
 
 -77 
 
 2.1598 
 
 .4630 
 
 .8484 
 
 1.3114 
 
 40.310 
 
 .78 
 
 2.1815 
 
 .4584 
 
 .8615 
 
 1.3199 
 
 40.746 
 
 -79 
 
 2.2034 
 
 .4538 
 
 .8748 
 
 1.3286 
 
 41.179 
 
 -80 
 
 2.2255 
 
 .4493 
 
 .8881 
 
 1.3374 
 
 41.608 
 
 -81 
 
 2.2479 
 
 .4449 
 
 .9015 
 
 1.3464 
 
 42.035 
 
 .82 
 
 2.2705 
 
 .4404 
 
 .9150 
 
 1.3555 
 
 42.460 
 
 -83 
 
 2.2933 
 
 .4360 
 
 .9286 
 
 1.3647 
 
 42.881 
 
 -84 
 
 2.3164 
 
 .4317 
 
 .9423 
 
 1.3740 
 
 43.299 
 
 -85 
 
 2.3396 
 
 .4274 
 
 .9561 
 
 1.3835 
 
 43.715 
 
 .86 
 
 2.3632 
 
 .4232 
 
 .9700 
 
 1.3932 
 
 44.128 
 
 .87 
 
 2.3869 
 
 .4190 
 
 .9840 
 
 1.4029 
 
 44.537 
 
 .88 
 
 2.4109 
 
 .4148 
 
 .9981 
 
 1.4128 
 
 44.944 
 
 .89 
 
 2.4351 
 
 .4107 
 
 1.0122 
 
 1.4229 
 
 45.348 
 
 .90 
 
 2.4596 
 
 .4066 
 
 1.0265 
 
 1.4331 
 
 45.750 
 
 .91 
 
 2.4843 
 
 .4025 
 
 1.0409 
 
 1.4434 
 
 46.148 
 
 -92 
 
 2.5093 
 
 .3985 
 
 1.0554 
 
 1.4539 
 
 46.544 
 
 -93 
 
 2.5345 
 
 .3946 
 
 1.0700 
 
 1.4645 
 
 46.936 
 
 -94 
 
 2.5600 
 
 .3906 
 
 1.0847 
 
 1.4753 
 
 47.326 
 
 -95 
 
 2.5857 
 
 .3867 
 
 1.0995 
 
 1.4862 
 
 47.713 
 
 -96 
 
 2.6117 
 
 .3829 
 
 1.1144 
 
 1.4973 
 
 48.097 
 
 .97 
 
 2.6379 
 
 .3791 
 
 1.1294 
 
 1.5085 
 
 48.478 
 
 .98 
 
 2.6645 
 
 .3753 
 
 1.1446 
 
 1.5199 
 
 48.857 
 
 .99 
 
 2.6912 
 
 .3716 
 
 1.1598 
 
 1.5314 
 
 49.232 
 
 1.00 
 
 2.7183 
 
 0.3679 
 
 1.1752 
 
 1.5431 
 
 49!60S 
 
 siiih X = tan gd x ; cosh a: = sec gd a; ; tanh x = sin gd x. 
 
126 
 
 TABLES. 
 
 Hyperbolic Functions. 
 
 X 
 
 I .si nil X 
 
 I cosh X 
 
 X 
 
 isinhx 
 
 I cosh X 
 
 X 
 
 isinhx 
 
 I cosh X 
 
 1.00 
 
 0.0701 
 
 0.1884 
 
 1.50 
 
 0.3282 
 
 0.3715 
 
 2.00 
 
 0.5595 
 
 0.5754 
 
 1.01 
 
 .0758 
 
 .1917 
 
 1.51 
 
 .3330 
 
 .3754 
 
 2.01 
 
 .5640 
 
 .5796 
 
 1.02 
 
 .0815 
 
 .1950 
 
 1.52 
 
 .3378 
 
 .3794 
 
 2.02 
 
 .5685 
 
 .5838 
 
 1.03 
 
 .0871 
 
 .1984 
 
 1.53 
 
 .3426 
 
 .3833 
 
 2.03 
 
 .5730 
 
 .5880 
 
 1.04 
 
 .0927 
 
 .2018 
 
 1.54 
 
 .3474 
 
 .3873 
 
 2.04 
 
 .5775 
 
 .5922 
 
 1.05 
 
 .0982 
 
 .2051 
 
 1.55 
 
 .3521 
 
 .3913 
 
 2.05 
 
 .5820 
 
 .5964 
 
 1.06 
 
 .1038 
 
 .2086 
 
 1.56 
 
 .3569 
 
 .3952 
 
 2.06 
 
 .5865 
 
 .6006 
 
 1.07 
 
 .1093 
 
 .2120 
 
 1.57 
 
 .3616 
 
 .3992 
 
 2.07 
 
 .5910 
 
 .6048 
 
 1.08 
 
 .1148 
 
 .2154 
 
 1.58 
 
 .3663 
 
 .4032 
 
 2.08 
 
 .5955 
 
 .6090 
 
 1.09 
 
 .1203 
 
 .2189 
 
 1.59 
 
 .3711 
 
 .4072 
 
 2.09 
 
 .6000 
 
 .6132 
 
 1.10 
 
 .1257 
 
 .2223 
 
 1.60 
 
 .3758 
 
 .4112 
 
 2.10 
 
 .6044 
 
 .6175 
 
 1.11 
 
 .1311 
 
 .2258 
 
 1.61 
 
 .3805 
 
 .4152 
 
 2.11 
 
 .6089 
 
 .6217 
 
 1.12 
 
 .1365 
 
 .2293 
 
 1.62 
 
 .3852 
 
 .4192 
 
 2.12 
 
 .6134 
 
 .6259 
 
 1.13 
 
 .1419 
 
 .2328 
 
 1.63 
 
 .3899 
 
 .4232 
 
 2.13 
 
 .6178 
 
 .6301 
 
 1.14 
 
 .1472 
 
 .2364 
 
 1.64 
 
 .3946 
 
 .4273 
 
 2.14 
 
 .6223 
 
 .6343 
 
 1.15 
 
 .1525 
 
 .2399 
 
 1.65 
 
 .3992 
 
 .4313 
 
 2.15 
 
 .6268 
 
 .6386 
 
 1.16 
 
 .1578 
 
 .2435 
 
 1.66 
 
 .4039 
 
 .4353 
 
 2.16 
 
 .6312 
 
 .6428 
 
 1.17 
 
 .1631 
 
 .2470 
 
 1.67 
 
 .4086 
 
 .4394 
 
 2.17 
 
 .6357 
 
 .6470 
 
 1.18 
 
 .1684 
 
 .2506 
 
 1.68 
 
 .4132 
 
 .4434 
 
 2.18 
 
 .6401 
 
 .6512 
 
 1.19 
 
 .1736 
 
 .2542 
 
 1.69 
 
 .4179 
 
 .4475 
 
 2.19 
 
 .6446 
 
 .6555 
 
 1.20 
 
 .1788 
 
 .2578 
 
 1.70 
 
 .4225 
 
 .4515 
 
 2.20 
 
 .6491 
 
 .6597 
 
 1.21 
 
 .1840 
 
 .2615 
 
 1.71 
 
 .4272 
 
 .4556 
 
 2.21 
 
 .6535 
 
 .6640 
 
 1.22 
 
 .1892 
 
 .2651 
 
 1.72 
 
 .4318 
 
 .4597 
 
 2.22 
 
 .6580 
 
 .6682 
 
 1.23 
 
 .1944 
 
 .2688 
 
 1.73 
 
 .4364 
 
 .4637 
 
 2.23 
 
 .6624 
 
 .6724 
 
 1.24 
 
 .1995 
 
 .2724 
 
 1.74 
 
 .4411 
 
 .4678 
 
 2.24 
 
 .6668 
 
 .6767 
 
 1.25 
 
 .2046 
 
 .2761 
 
 1.75 
 
 .4457 
 
 .4719 
 
 2.25 
 
 .6713 
 
 .6809 
 
 1.26 
 
 .2098 
 
 .2798 
 
 1.76 
 
 .4503 
 
 .4760 
 
 2.26 
 
 .6757 
 
 .6852 
 
 1.27 
 
 .2148 
 
 .2835 
 
 1.77 
 
 .4549 
 
 .4801 
 
 2.27 
 
 .6802 
 
 .6894 
 
 1.28 
 
 .2199 
 
 .2872 
 
 1.78 
 
 .4595 
 
 .4842 
 
 2.28 
 
 .6846 
 
 .6937 
 
 1.29 
 
 .2250 
 
 .2909 
 
 1.79 
 
 .4641 
 
 .4883 
 
 2.29 
 
 .6890 
 
 .6979 
 
 1.30 
 
 .2300 
 
 .2947 
 
 1.80 
 
 .4687 
 
 .4924 
 
 2.30 
 
 .6935 
 
 .7022 
 
 1.31 
 
 .2351 
 
 .2984 
 
 LSI 
 
 .4733 
 
 .4965 
 
 2.31 
 
 .6979 
 
 .7064 
 
 1.32 
 
 .2401 
 
 .3022 
 
 1.82 
 
 .4778 
 
 .5006 
 
 2.32 
 
 .7023 
 
 .7107 
 
 1.33 
 
 .2451 
 
 .3059 
 
 1.83 
 
 .4824 
 
 .5048 
 
 2.33 
 
 .7067 
 
 .7150 
 
 1.34 
 
 .2501 
 
 .3097 
 
 1.84 
 
 .4870 
 
 .5089 
 
 2.34 
 
 .7112 
 
 .7192 
 
 1.35 
 
 .2551 
 
 .3135 
 
 1.85 
 
 .4915 
 
 .5130 
 
 2.35 
 
 .7156 
 
 .7235 
 
 1.36 
 
 .2600 
 
 .3173 
 
 1.86 
 
 .4961 
 
 .5172 
 
 2.36 
 
 .7200 
 
 .7278 
 
 1.37 
 
 .2650 
 
 .3211 
 
 1.87 
 
 .5007 
 
 .5213 
 
 2.37 
 
 .7244 
 
 .7320 
 
 1.38 
 
 .2699 
 
 .3249 
 
 1.88 
 
 .5052 
 
 .5254 
 
 2.38 
 
 .7289 
 
 .7363 
 
 1.39 
 
 .2748 
 
 .3288 
 
 1.89 
 
 .5098 
 
 .5296 
 
 2.38 
 
 .7333 
 
 .7406 
 
 1.40 
 
 .2797 
 
 .3326 
 
 1.90 
 
 .5143 
 
 .5337 
 
 2.40 
 
 .7377 
 
 .7448 
 
 1.41 
 
 .2846 
 
 .3365 
 
 1.91 
 
 .5188 
 
 .5379 
 
 2.41 
 
 .7421 
 
 .7491 
 
 1.42 
 
 .2895 
 
 .3403 
 
 1.92 
 
 .5234 
 
 .5421 
 
 2.42 
 
 .7465 
 
 .7534 
 
 1.43 
 
 .2944 
 
 .3442 
 
 1.93 
 
 .5279 
 
 .5462 
 
 2.43 
 
 .7509 
 
 .7577 
 
 1.44 
 
 .2993 
 
 -3481 
 
 1.94 
 
 .5324 
 
 .5504 
 
 2.44 
 
 .7553 
 
 .7619 
 
 1.45 
 
 .3041 
 
 .3520 
 
 1.95 
 
 .5370 
 
 .5545 
 
 2.45 
 
 .7597 
 
 .7662 
 
 1.46 
 
 .3090 
 
 .3559 
 
 1.96 
 
 .5415 
 
 .5687 
 
 2.46 
 
 .7642 
 
 .7705 
 
 1.47 
 
 .3138 
 
 .3598 
 
 1.97 
 
 .5460 
 
 .5629 
 
 2.47 
 
 .7686 
 
 .7748 
 
 1.48 
 
 .3186 
 
 .3637 
 
 1.98 
 
 .5505 
 
 .5671 
 
 2.48 
 
 .7730 
 
 .7791 
 
 1.49 
 
 .3234 
 
 .3676 
 
 1.99 
 
 .5550 
 
 .5713 
 
 2.49 
 
 .7774 
 
 .7833 
 
 1.50 
 
 0.3282 
 
 0.3715 
 
 2.00 
 
 0.5595 
 
 0.5754 
 
 2.50 
 
 0.7818 
 
 0.7876 
 
TABLES. 
 
 127 
 
 Hyperbolic Functions. 
 
 X 
 
 I sinh X 
 
 I cosh X 
 
 X 
 
 isinhx 
 
 I cosh X 
 
 X 
 
 I sinh X 
 
 I cosh X 
 
 2.50 
 
 0.7S1S 
 
 0.7876 
 
 2.1 S 
 
 0.8915 
 
 0.8951 
 
 3.0 
 
 1.0008 
 
 1.0029 
 
 2.51 
 
 .7862 
 
 .7919 
 
 2.76 
 
 .8959 
 
 .8994 
 
 3.1 
 
 1.0444 
 
 1.0462 
 
 2.52 
 
 .7906 
 
 .7962 
 
 2.77 
 
 .9003 
 
 .9037 
 
 3.2 
 
 1.0880 
 
 1.0894 
 
 2.53 
 
 .7950 
 
 .8005 
 
 2.78 
 
 .9046 
 
 .9080 
 
 3.3 
 
 1.1316 
 
 1.1327 
 
 2.54 
 
 .7994 
 
 .8048 
 
 2.79 
 
 .9090 
 
 .9123 
 
 3.4 
 
 1.1751 
 
 1.1761 
 
 2.55 
 
 .8038 
 
 .8091 
 
 2.80 
 
 .9134 
 
 .9166 
 
 3.5 
 
 1.2186 
 
 1.2194 
 
 2.56 
 
 .8082 
 
 .8134 
 
 2.81 
 
 .9178 
 
 .9209 
 
 3.6 
 
 1.2621 
 
 1.2628 
 
 2.57 
 
 .8126 
 
 .8176 
 
 2.82 
 
 .9221 
 
 .9252 
 
 3.7 
 
 1.3056 
 
 1.3061 
 
 2.58 
 
 .8169 
 
 .8219 
 
 2.83 
 
 .9265 
 
 .9295 
 
 3.8 
 
 1.3491 
 
 1.3495 
 
 2.59 
 
 .8213 
 
 .8262 
 
 2.84 
 
 .9309 
 
 .9338 
 
 3.9 
 
 1.3925 
 
 1.3929 
 
 2.60 
 
 .8257 
 
 .8305 
 
 2.85 
 
 .9353 
 
 .9382 
 
 4.0 
 
 1.4360 
 
 1.4363 
 
 2.61 
 
 .8301 
 
 .8348 
 
 2.86 
 
 .9396 
 
 .9425 
 
 4.1 
 
 1.4795 
 
 1.4797 
 
 2.62 
 
 .8345 
 
 .8391 
 
 2.87 
 
 .9440 
 
 .9468 
 
 4.2 
 
 1.5229 
 
 1.5231 
 
 2.63 
 
 .8389 
 
 .8434 
 
 2.88 
 
 .9484 
 
 .9511 
 
 4.3 
 
 1.5664 
 
 1.5665 
 
 2.64 
 
 .8433 
 
 .8477 
 
 2.89 
 
 .9527 
 
 .9554 
 
 4.4 
 
 1.6098 
 
 1.6099 
 
 2.65 
 
 .8477 
 
 .8520 
 
 2 90 
 
 .9571 
 
 .9597 
 
 4.5 
 
 1.6532 
 
 1.6533 
 
 2.66 
 
 .8521 
 
 .8563 
 
 2.91 
 
 .9615 
 
 .9641 
 
 4.6 
 
 1.6967 
 
 1.6968 
 
 2.67 
 
 .8564 
 
 .8606 
 
 2.92 
 
 .9658 
 
 .9684 
 
 4.7 
 
 1.740] 
 
 1.7402 
 
 2.68 
 
 .8608 
 
 .8649 
 
 2.93 
 
 .9702 
 
 .9727 
 
 4.8 
 
 1.7836 
 
 1.7836 
 
 2.69 
 
 .8652 
 
 .8692 
 
 2.94 
 
 .9746 
 
 .9770 
 
 4.9 
 
 1.8270 
 
 1.8270 
 
 2.70 
 
 .8696 
 
 .8735 
 
 2.95 
 
 .9789 
 
 .9813 
 
 5.0 
 
 1.8704 
 
 1.8705 
 
 2.71 
 
 .8740 
 
 .8778 
 
 2.96 
 
 .9833 
 
 .9856 
 
 6.0 
 
 2.3047 
 
 2.3047 
 
 2.72 
 
 .8784 
 
 .8821 
 
 2.97 
 
 .9877 
 
 .9900 
 
 7.0 
 
 2.7390 
 
 2.7390 
 
 2.73 
 
 .8827 
 
 .8864 
 
 2.98 
 
 .9920 
 
 .9943 
 
 8.0 
 
 3.1733 
 
 3.1733 
 
 2.74 
 
 .8871 
 
 .8907 
 
 2.99 
 
 .9964 
 
 .9986 
 
 9.0 
 
 3.6076 
 
 3.6076 
 
 2.75 
 
 0.8915 
 
 0.8951 
 
 3.00 
 
 1.0008 
 
 1.0029 
 
 10.0 
 
 4.0419 
 
 4.0419 
 
 For values of x greater than 7.0, we may write, to five places of deci- 
 mals at least, 
 
 logio sinh X = logio cosh x = log i e^ = x (0.4342945) + 1.6989700. 
 
 The Values of e-x^ for Certain Values of x. 
 
 X 
 
 e-^ 
 
 X 
 
 e-^ 
 
 X 
 
 Q-X 
 
 X 
 
 Q-X 
 
 1/10 
 
 0.90484 
 
 8/10 
 
 0.44933 
 
 18/10 
 
 0.16530 
 
 5 
 
 0.00674 
 
 1/8 
 
 0.88250 
 
 9/10 
 
 0.40657 
 
 2 
 
 0.13534 
 
 11/2 
 
 0.00409 
 
 1/6 
 
 0.84648 
 
 1 
 
 0.36788 
 
 9/4 
 
 0.10540 
 
 6 
 
 0.00248 
 
 2/10 
 
 0.81873 
 
 11/10 
 
 0.33287 
 
 5/2 
 
 0.08209 
 
 13/2 
 
 0.00150 
 
 1/4 
 
 0.77880 
 
 9/8 
 
 0.32465 
 
 8/3 
 
 0.06948 
 
 7 
 
 0.00091 
 
 3/10 
 
 0.74082 
 
 12/10 
 
 0.30119 
 
 3 
 
 0.04979 
 
 15/2 
 
 0.00055 
 
 1/3 
 
 0.71653 
 
 5/4 
 
 0.28650 
 
 25/8 
 
 0.04394 
 
 8 
 
 0.00034 
 
 4/10 
 
 0.67032 
 
 13/10 
 
 0.27253 
 
 16/5 
 
 0.04076 
 
 9 
 
 0.00012 
 
 5/10 
 
 0.60653 
 
 4/3 
 
 0.26360 
 
 18/5 
 
 0.02732 
 
 10 
 
 0.00004 
 
 6/10 
 
 0.54881 
 
 14/10 
 
 0.24660 
 
 4 
 
 0.01832 
 
 11 
 
 0.00002 
 
 2/3 
 
 0.51342 
 
 3/2 
 
 0.22313 
 
 25/6 
 
 0.01550 
 
 12 
 
 0.00001 
 
 7/10 
 
 0.49659 
 
 16/10 
 
 0.20190 
 
 9/2 
 
 0.01111 
 
 13 
 
 0.00000 
 
128 
 
 TABLES. 
 
 The Common Logarithms of e^ and e-«. 
 
 
 « 
 
 logioe^ 
 
 logioe-^ 
 
 
 0.00001 
 
 0.0000043429 
 
 1.9999956571 
 
 
 0.00002 
 
 0.0000086859 
 
 1.9999913141 
 
 
 0.00003 
 
 0.0000130288 
 
 1.9999869712 
 
 
 0.00004 
 
 0.0000173718 
 
 1.9999826282 
 
 
 0.00005 
 
 0.0000217147 
 
 1.9999782853 
 
 
 0.00006 
 
 0.0000260577 
 
 1.9999739423 
 
 
 0.00007 
 
 0.0000304006 
 
 1.9999695994 • 
 
 
 0.00008 
 
 0.0000347436 
 
 1.9999652564 
 
 
 0.00009 
 
 0.0000390865 
 
 1.9999609135 
 
 
 0.00010 
 
 0.0000434294 
 
 1.9999565706 
 
 
 0.00020 
 
 0.0000868589 
 
 1.9999131411 
 
 
 0.00030 
 
 0.0001302883 
 
 1.9998697117 
 
 
 0.00040 
 
 0.0001737178 
 
 1.9998262822 
 
 
 0.00050 
 
 0.0002171472 
 
 1.9997828528 
 
 
 0.00060 
 
 0.0002605767 
 
 1.9997394233 
 
 
 0.00070 
 
 0.0003040061 
 
 1.9996959939 
 
 
 0.00080 
 
 0.0003474356 
 
 1.9996525644 
 
 
 0.00090 
 
 0.0003908650 
 
 1.9996091350 
 
 
 0.00100 
 
 0.0004342945 
 
 1.9995657055 
 
 
 0.00200 
 
 0.0008685890 
 
 1.9991314110 
 
 
 0.00300 
 
 0.0013028834 
 
 1.9986971166 
 
 
 0.00400 
 
 0.0017371779 
 
 1.9982628221 
 
 
 0.00500 
 
 0.0021714724 
 
 1.9978285276 
 
 
 0.00600 
 
 0.0026057669 
 
 1.9973942331 
 
 
 0.00700 
 
 0.0030400614 
 
 1.9969599386 
 
 
 0.00800 
 
 0.0034743559 
 
 1.9965256441 
 
 
 0.00900 
 
 0.0039086503 
 
 1.9960913497 
 
 
 0.01000 
 
 0.0043429448 
 
 1.9956570552 
 
 
 0.02000 
 
 0.0086858896 
 
 1.9913141 ICH- 
 
 
 0.03000 
 
 0.0130288345 
 
 1.9869711655 
 
 
 0.04000 
 
 0.0173717793 
 
 T.9826282207 
 
 
 0.05000 
 
 0.0217147241 
 
 1.9782852759 
 
 
 0.06000 
 
 0.0260576689 
 
 1.9739423311 
 
 
 0.07000 
 
 0.0304006137 
 
 1.9695993863 
 
 
TABLES. 
 
 129 
 
 
 X 
 
 logio e* 
 
 logio e-" 
 
 
 
 0.08000 
 
 0.0347435586 
 
 1.9652564414 
 
 
 
 0.09000 
 
 0.0390865034 
 
 1.9609134966 
 
 
 
 0.10000 
 
 0.0434294482 
 
 1.9565705518 
 
 
 
 0.20000 
 
 0.0868588964 
 
 1.9131411036 
 
 
 
 0.30000 
 
 0.1302883446 
 
 1.8697116554 
 
 
 
 0.40000 
 
 0.1737177928 
 
 1.8262822072 
 
 
 
 0.50000 
 
 02171472410 
 
 1.7828527590 
 
 
 
 0.60000 
 
 0.2605766891 
 
 1.7394233109 
 
 
 
 0.70000 
 
 0.3040061373 
 
 1.6959938627 
 
 
 
 0.80000 
 
 0.3474355855 
 
 1.6525644145 
 
 
 
 090000 
 
 0.3908650337 
 
 1.6091349663 
 
 
 
 1.00000 
 
 0.4342944819 
 
 1.5657055181 
 
 
 
 2.00000 
 
 0.8685889638 
 
 1.1314110362 
 
 
 
 3.00000 
 
 1.3028834457 
 
 2.6971165543 
 
 
 
 4.00000 
 
 1.7371779276 
 
 2.2628220724 
 
 
 
 5.00000 
 
 2.1714724095 
 
 3.8285275905 
 
 
 
 6.00000 
 
 2.6057668914 
 
 3.3942331086 
 
 
 
 7.00000 
 
 3.0400613733 
 
 4.9599386267 
 
 
 
 8.00000 
 
 3.4743558552 
 
 4.5256441448 
 
 
 
 9.00000 
 
 3.9086503371 
 
 4.0913496629 
 
 
 
 10.00000 
 
 4.3429448190 
 
 5.6570551810 
 
 
 
 20.00000 
 
 8.6858896381 
 
 9.3141103619 
 
 
 
 30.00000 
 
 13.0288344571 
 
 14.9711655429 ^ 
 
 
 
 40.00000 
 
 17.3717792761 
 
 18.6282207239 
 
 
 
 50.00000 
 
 21.7147240952 
 
 22.2852759048 
 
 
 
 60.00000 
 
 26.0576689142 
 
 27.9423310858 
 
 
 
 70.00000 
 
 30.4006137332 
 
 31.5993862668 
 
 
 
 80.00000 
 
 34.7435585523 
 
 35.2564414477 
 
 
 
 90.00000 
 
 39.0865033713 
 
 40.9134966287 
 
 
 
 100.00000 
 
 43.4294481903 
 
 44.5705518097 
 
 
 
 200.00000 
 
 86.8588963807 
 
 87.1411036193 
 
 
 
 300.00000 
 
 130.2883445710 
 
 131.7116554290 
 
 
 
 400.00000 
 
 173.7177927613 
 
 174.2822072387 
 
 
 
 500.00000 
 
 217.1472409516 
 
 218.8527590tS4 
 
 
 Note •- log e^ + V = log e* + log &>. Thus, log giis-iirs - 49.139465 ISa 
 
130 
 
 TABLES. 
 
 Five-Place Natural Logarithms. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 1.00 
 
 0.0 0000 
 
 0100 
 
 0200 
 
 0300 
 
 0399 
 
 0499 
 
 0598 
 
 0698 
 
 0797 
 
 0896 
 
 100-99 
 
 1.01 
 
 0.0 0995 
 
 1094 
 
 1193 
 
 1292 
 
 1390 
 
 1489 
 
 1587 
 
 1686 
 
 1784 
 
 1882 
 
 99-98 
 
 1.02 
 
 0.0 1980 
 
 2078 
 
 2176 
 
 2274 
 
 2372 
 
 2469 
 
 2567 
 
 2664 
 
 2762 
 
 2859 
 
 98-97 
 
 1.03 
 
 0.0 2956 
 
 3053 
 
 3150 
 
 3247 
 
 3343 
 
 3440 
 
 3537 
 
 3633 
 
 3730 
 
 3826 
 
 97-96 
 
 1.04 
 
 0.0 3922 
 
 4018 
 
 4114 
 
 4210 
 
 4306 
 
 4402 
 
 4497 
 
 4593 
 
 4688 
 
 4784 
 
 96-95 
 
 1.05 
 
 0.0 4879 
 
 4974 
 
 5069 
 
 5164 
 
 5259 
 
 5354 
 
 5449 
 
 5543 
 
 5638 
 
 5733 
 
 95-94 
 
 1.06 
 
 0.0 5827 
 
 5921 
 
 6015 
 
 6110 
 
 6204 
 
 6297 
 
 6391 
 
 6485 
 
 6579 
 
 6672 
 
 94 
 
 1.07 
 
 0.0 6766 
 
 6859 
 
 6953 
 
 7046 
 
 7139 
 
 7232 
 
 7325 
 
 7418 
 
 7511 
 
 7603 
 
 93 
 
 1.08 
 
 0.0 7696 
 
 7789 
 
 7881 
 
 7973 
 
 8066 
 
 8158 
 
 8250 
 
 8342 
 
 8434 
 
 8526 
 
 93-92 
 
 1.09 
 
 0.0 8618 
 
 8709 
 
 8801 
 
 8893 
 
 8984 
 
 9075 
 
 9167 
 
 9258 
 
 9349 
 
 9430 
 
 92-91 
 
 1.10 
 
 0.0 9531 
 
 9622 
 
 9713 
 
 9803 
 
 9894 
 
 9985 
 
 *0075 
 
 0165 
 
 0256 
 
 0346 
 
 91-90 
 
 1.11 
 
 0.1 0436 
 
 0526 
 
 0616 
 
 0706 
 
 0796 
 
 0885 
 
 0975 
 
 1065 
 
 1154 
 
 1244 
 
 90-89 
 
 1.12 
 
 0.1 1333 
 
 1422 
 
 1511 
 
 1600 
 
 1689 
 
 1778 
 
 1867 
 
 1956 
 
 2045 
 
 2133 
 
 89 
 
 1.13 
 
 0.1 2222 
 
 2310 
 
 2399 
 
 2487 
 
 2575 
 
 2663 
 
 2751 
 
 2839 
 
 2927 
 
 3015 
 
 88 
 
 1.14 
 
 0.1 3103 
 
 3191 
 
 3278 
 
 3366 
 
 3453 
 
 3540 
 
 3628 
 
 3715 
 
 3802 
 
 3889 
 
 88-87 
 
 1.15 
 
 0.1 3976 
 
 4063 
 
 4150 
 
 4237 
 
 4323 
 
 4410 
 
 4497 
 
 4583 
 
 4669 
 
 4756 
 
 87-86 
 
 1.16 
 
 0.1 4842 
 
 4928 
 
 5014 
 
 5100 
 
 5186 
 
 5272 
 
 5358 
 
 5444 
 
 5529 
 
 5615 
 
 86 
 
 1.17 
 
 0.1 5700 
 
 5786 
 
 5871 
 
 5956 
 
 6042 
 
 6127 
 
 6212 
 
 6297 
 
 6382 
 
 6467 
 
 85 
 
 1.18 
 
 0.16551 
 
 6636 
 
 6721 
 
 6805 
 
 6890 
 
 6974 
 
 7059 
 
 7143 
 
 7227 
 
 7311 
 
 85-84 
 
 1.19 
 
 0.1 7395 
 
 7479 
 
 7563 
 
 7647 
 
 7731 
 
 7815 
 
 7898 
 
 7982 
 
 8065 
 
 8149 
 
 84-83 
 
 1.20 
 
 0.1 8232 
 
 8315 
 
 8399 
 
 8482 
 
 8565 
 
 8648 
 
 8731 
 
 8814 
 
 8897 
 
 9979 
 
 83 
 
 1.21 
 
 0.1 9062 
 
 9145 
 
 9227 
 
 9310 
 
 9392 
 
 9474 
 
 9557 
 
 9639 
 
 9721 
 
 9803 
 
 83-82 
 
 1.22 
 
 0.1 9885 
 
 9967 *0049 
 
 0131 
 
 0212 
 
 0294 
 
 0376 
 
 0457 
 
 0539 
 
 0620 
 
 82-81 
 
 1.23 
 
 0.2 0701 
 
 0783 
 
 0864 
 
 0945 
 
 1026 
 
 1107 
 
 1188 
 
 1269 
 
 1350 
 
 1430 
 
 81 
 
 1.24 
 
 0.21511 
 
 1592 
 
 1672 
 
 1753 
 
 1833 
 
 1914 
 
 1994 
 
 2074 
 
 2154 
 
 2234 
 
 81-80 
 
 1.25 
 
 0.2 2314 
 
 2394 
 
 2474 
 
 2554 
 
 2634 
 
 2714 
 
 2793 
 
 2873 
 
 2952 
 
 3032 
 
 80-79 
 
 1.26 
 
 0.2 3111 
 
 3191 
 
 3270 
 
 3349 
 
 3428 
 
 3507 
 
 3586 
 
 3665 
 
 3744 
 
 3823 
 
 79 
 
 1.27 
 
 0.2 3902 
 
 3980 
 
 4059 
 
 4138 
 
 4216 
 
 4295 
 
 4373 
 
 4451 
 
 4530 
 
 4608 
 
 79-78 
 
 1.28 
 
 0.2 4686 
 
 4764 
 
 4842 
 
 4920 
 
 4998 
 
 5076 
 
 5154 
 
 5231 
 
 5309 
 
 5387 
 
 78 
 
 1.29 
 
 0.2 5464 
 
 5542 
 
 5619 
 
 5697 
 
 5774 
 
 5811 
 
 5928 
 
 6005 
 
 6082 
 
 6159 
 
 77 
 
 1.30 
 
 0.2 6236 
 
 6313 
 
 6390 
 
 6467 
 
 6544 
 
 6620 
 
 6697 
 
 6773 
 
 6850 
 
 6926 
 
 77-76 
 
 1.31 
 
 0.2 7003 
 
 7079 
 
 7155 
 
 7231 
 
 7308 
 
 7384 
 
 7460 
 
 7536 
 
 7612 
 
 7687 
 
 76 
 
 1.32 
 
 0.2 7763 
 
 7839 
 
 7915 
 
 7990 
 
 8066 
 
 8141 
 
 8217 
 
 8292 
 
 8367 
 
 8443 
 
 76-75 
 
 1.33 
 
 0.2 8518 
 
 8593 
 
 8668 
 
 8743 
 
 8818 
 
 8893 
 
 8968 
 
 9043 
 
 9118 
 
 9192 
 
 75 
 
 1.34 
 
 0.2 9267 
 
 9342 
 
 9416 
 
 9491 
 
 9565 
 
 9639 
 
 9714 
 
 9788 
 
 9862 
 
 9936 
 
 75-74 
 
 1.35 
 
 0.3 0010 
 
 0085 
 
 0158 
 
 0232 
 
 0306 
 
 0380 
 
 0454 
 
 0528 
 
 0601 
 
 0675 
 
 74 
 
 1.36 
 
 0.3 0748 
 
 0822 
 
 0895 
 
 0969 
 
 1042 
 
 1115 
 
 1189 
 
 1262 
 
 1335 
 
 1408 
 
 74-73 
 
 1.37 
 
 0.3 1481 
 
 1554 
 
 1627 
 
 1700 
 
 1773 
 
 1845 
 
 1918 
 
 1991 
 
 2063 
 
 2136 
 
 73-72 
 
 1.38 
 
 0.3 2208 
 
 2281 
 
 2353 
 
 2426 
 
 2498 
 
 2570 
 
 2642 
 
 2714 
 
 2786 
 
 2858 
 
 72 
 
 1.39 
 
 0.3 2930 
 
 3002 
 
 3074 
 
 3146 
 
 3218 
 
 3289 
 
 3361 
 
 3433 
 
 3504 
 
 3576 
 
 72-71 
 
 1.40 
 
 0.3 3647 
 
 3719 
 
 3790 
 
 3861 
 
 3933 
 
 4004 
 
 4075 
 
 4146 
 
 4217 
 
 4288 
 
 71 
 
 1.41 
 
 0.3 4359 
 
 4430 
 
 4501 
 
 4572 
 
 4642 
 
 4713 
 
 4784 
 
 4854 
 
 4925 
 
 4995 
 
 71-70 
 
 1.42 
 
 0.3 5066 
 
 5136 
 
 5206 
 
 5277 
 
 5347 
 
 5417 
 
 5487 
 
 5557 
 
 5677 
 
 5697 
 
 70 
 
 1.43 
 
 0.3 5767 
 
 5837 
 
 5907 
 
 5977 
 
 6047 
 
 6116 
 
 6186 
 
 6256 
 
 6335 
 
 6395 
 
 70-69 
 
 1.44 
 
 0.3 6464 
 
 6534 
 
 6603 
 
 6672 
 
 6742 
 
 6811 
 
 6880 
 
 6949 
 
 7018 
 
 7087 
 
 69 
 
 1.45 
 
 0.3 7156 
 
 7225 
 
 7294 
 
 7363 
 
 7432 
 
 7501 
 
 7569 
 
 7638 
 
 7707 
 
 7775 
 
 69 
 
 1.46 
 
 0.3 7844 
 
 7912 
 
 7981 
 
 8049 
 
 8117 
 
 8186 
 
 8254 
 
 8322 
 
 8390 
 
 8458 
 
 68 
 
 1.47 
 
 0.3 8526 
 
 8594 
 
 8662 
 
 8730 
 
 8798 
 
 8866 
 
 8934 
 
 9001 
 
 9069 
 
 9137 
 
 68 
 
 1.48 
 
 0.3 9204 
 
 9272 
 
 9339 
 
 9407 
 
 9474 
 
 9541 
 
 9609 
 
 9676 
 
 9743 
 
 9810 
 
 68-67 
 
 1.49 
 
 0.3 9878 
 
 9945 
 
 *0012 
 
 0079 
 
 0146 
 
 0213 
 
 0279 
 
 0346 
 
 0413 
 
 0480 
 
 67 
 
 1.50 
 
 0.4 0547 
 
 0613 
 
 0680 
 
 0746 
 
 0813 
 
 0879 
 
 0946 
 
 1012 
 
 1078 
 
 1145 
 
 67-€6 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
TABLES. 
 
 131 
 
 Five-Place Natural Logarithms. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 1.50 
 
 0.4 0547 
 
 0613 
 
 0680 
 
 0746 
 
 0813 
 
 0879 
 
 0946 
 
 1012 
 
 1078 
 
 1145 
 
 67-66 
 
 1.51 
 
 0.41211 
 
 1277 
 
 1343 
 
 1409 
 
 1476 
 
 1542 
 
 1608 
 
 1673 
 
 1739 
 
 1805 
 
 66 
 
 1.52 
 
 0.4 1871 
 
 1937 
 
 2003 
 
 2068 
 
 2134 
 
 2199 
 
 2265 
 
 2331 
 
 2396 
 
 2461 
 
 66-65 
 
 1.53 
 
 0.4 2527 
 
 2592 
 
 2657 
 
 2723 
 
 2788 
 
 2853 
 
 2918 
 
 2983 
 
 3048 
 
 3113 
 
 65 
 
 1.54 
 
 0.4 3178 
 
 3243 
 
 3308 
 
 3373 
 
 3438 
 
 3502 
 
 3567 
 
 3632 
 
 3696 
 
 3761 
 
 65-64 
 
 1.55 
 
 0.4 3825 
 
 3890 
 
 3954 
 
 4019 
 
 4083 
 
 4148 
 
 4212 
 
 4276 
 
 4340 
 
 4404 
 
 64 
 
 1.56 
 
 0.4 4469 
 
 4533 
 
 4597 
 
 4661 
 
 4725 
 
 4789 
 
 4852 
 
 4916 
 
 4980 
 
 5044 
 
 64 
 
 1.57 
 
 0.4 5108 
 
 5171 
 
 5235 
 
 5298 
 
 5362 
 
 5426 
 
 5489 
 
 5552 
 
 5616 
 
 5679 
 
 64-63 
 
 1.58 
 
 0.4 5742 
 
 5S06 
 
 5869 
 
 5932 
 
 5995 
 
 6058 
 
 6122 
 
 6185 
 
 6248 
 
 6310 
 
 63 
 
 1.59 
 
 0.4 6373 
 
 6436 
 
 6499 
 
 6562 
 
 6625 
 
 6687 
 
 6750 
 
 6813 
 
 6875 
 
 6938 
 
 63 
 
 1.60 
 
 0.4 7000 
 
 7063 
 
 7125 
 
 7188 
 
 7250 
 
 7312 
 
 7375 
 
 7437 
 
 7499 
 
 7561 
 
 62 
 
 1.61 
 
 0.4 7623 
 
 7686 
 
 7748 
 
 7810 
 
 7872 
 
 7933 
 
 7995 
 
 8057 
 
 8119 
 
 8181 
 
 63 
 
 1.62 
 
 0.4 8243 
 
 8304 
 
 8366 
 
 8428 
 
 8489 
 
 8551 
 
 8612 
 
 8674 
 
 8735 
 
 8797 
 
 62-61 
 
 1.63 
 
 0.4 8858 
 
 8919 
 
 8981 
 
 9042 
 
 9103 
 
 9164 
 
 9225 
 
 9287 
 
 9348 
 
 9409 
 
 61 
 
 1.64 
 
 0.4 9470 
 
 9531 
 
 9592 
 
 9652 
 
 9713 
 
 9774 
 
 9835 
 
 9896 
 
 9956 *0017 
 
 61 
 
 1.65 
 
 0.5 0078 
 
 013S 
 
 0199 
 
 0259 
 
 0320 
 
 0380 
 
 0441 
 
 0501 
 
 0561 
 
 0622 
 
 61-60 
 
 1.66 
 
 ■ 0.5 0682 
 
 0742 
 
 0802 
 
 0862 
 
 0922 
 
 0983 
 
 1043 
 
 1103 
 
 1163 
 
 1222 
 
 60 
 
 1.67 
 
 0.5 1282 
 
 1342 
 
 1402 
 
 1462 
 
 1522 
 
 1581 
 
 1641 
 
 1701 
 
 1760 
 
 1820 
 
 60 
 
 1.68 
 
 0.5 1879 
 
 1939 
 
 1998 
 
 2058 
 
 2117 
 
 2177 
 
 2236 
 
 2295 
 
 2354 
 
 2414 
 
 60-59 
 
 1.69 
 
 0.5 2473 
 
 2532 
 
 2591 
 
 2650 
 
 2709 
 
 2768 
 
 2827 
 
 2886 
 
 2945 
 
 3004 
 
 59 
 
 1.70 
 
 0.5 3063 
 
 3122 
 
 3180 
 
 3239 
 
 3298 
 
 3357 
 
 3415 
 
 3474 
 
 3532 
 
 3591 
 
 59-58 
 
 1.71 
 
 0.5 3649 
 
 3708 
 
 3766 
 
 3825 
 
 3883 
 
 3941 
 
 4000 
 
 4058 
 
 4116 
 
 4174 
 
 58 
 
 1.72 
 
 0.5 4232 
 
 4291 
 
 4349 
 
 4407 
 
 4465 
 
 4523 
 
 4581 
 
 4639 
 
 4696 
 
 4754 
 
 58 
 
 1.73 
 
 0.5 4812 
 
 4870 
 
 4928 
 
 4985 
 
 5043 
 
 5101 
 
 5158 
 
 5216 
 
 5274 
 
 5331 
 
 58-57 
 
 1.74 
 
 0.5 5389 
 
 5446 
 
 5503 
 
 5561 
 
 5618 
 
 5675 
 
 5/oj 
 
 5790 
 
 5847 
 
 5904 
 
 57 
 
 1.75 
 
 0.5 5962 
 
 6019 
 
 6076 
 
 6133 
 
 6190 
 
 6247 
 
 6304 
 
 6361 
 
 6418 
 
 6475 
 
 57 
 
 1.76 
 
 0.5 6531 
 
 6588 
 
 6645 
 
 6702 
 
 6758 
 
 6815 
 
 6872 
 
 6928 
 
 6985 
 
 7041 
 
 57 
 
 1.77 
 
 0.5 7098 
 
 7154 
 
 7211 
 
 7267 
 
 7324 
 
 7380 
 
 7436 
 
 7493 
 
 7549 
 
 7605 
 
 56 
 
 1.78 
 
 0.5 7661 
 
 7718 
 
 7774 
 
 7830 
 
 7886 
 
 7942 
 
 7998 
 
 8054 
 
 8110 
 
 8166 
 
 56 
 
 1.79 
 
 0.5 8222 
 
 8277 
 
 8333 
 
 8389 
 
 8445 
 
 8501 
 
 8556 
 
 8612 
 
 8667 
 
 8723 
 
 56 
 
 1.80 
 
 0.5 8779 
 
 8834 
 
 8890 
 
 8945 
 
 9001 
 
 9056 
 
 9111 
 
 9167 
 
 9222 
 
 9277 
 
 56-55 
 
 1.81 
 
 0.5 9333 
 
 9388 
 
 9443 
 
 9498 
 
 9553 
 
 9609 
 
 9664 
 
 9719 
 
 9774 
 
 9829 
 
 55 
 
 1.82 
 
 0.5 9884 
 
 9939 
 
 9993 
 
 *004S 
 
 0103 
 
 0158 
 
 0213 
 
 0268 
 
 0322 
 
 0377 
 
 55 
 
 1.83 
 
 0.6 0432 
 
 0486 
 
 0541 
 
 0595 
 
 0650 
 
 0704 
 
 0759 
 
 0813 
 
 0868 
 
 0922 
 
 55-54 
 
 1.84 
 
 0.6 0977 
 
 1031 
 
 1085 
 
 1139 
 
 1194 
 
 1248 
 
 1302 
 
 1356 
 
 1410 
 
 1464 
 
 54 
 
 1.85 
 
 0.61519 
 
 1573 
 
 1627 
 
 1681 
 
 1735 
 
 1788 
 
 1842 
 
 1896 
 
 1950 
 
 2004 
 
 54 
 
 1.86 
 
 0.6 2058 
 
 2111 
 
 2165 
 
 2219 
 
 2272 
 
 2326 
 
 2380 
 
 2433 
 
 2487 
 
 2540 
 
 54-53 
 
 1.87 
 
 0.6 2594 
 
 2647 
 
 2701 
 
 2754 
 
 2808 
 
 2861 
 
 2914 
 
 2967 
 
 3021 
 
 3074 
 
 53 
 
 1.88 
 
 0.6 3127 
 
 3180 
 
 3234 
 
 3287 
 
 3340 
 
 3393 
 
 3446 
 
 3499 
 
 3552 
 
 3605 
 
 53 
 
 1.89 
 
 0.6 3658 
 
 3711 
 
 3763 
 
 3816 
 
 3869 
 
 3922 
 
 3975 
 
 4027 
 
 4080 
 
 4133 
 
 53 
 
 1.90 
 
 0.6 4185 
 
 4238 
 
 4291 
 
 4343 
 
 4396 
 
 4448 
 
 4501 
 
 4553 
 
 4606 
 
 4658 
 
 53-52 
 
 1.91 
 
 0.6 4710 
 
 4763 
 
 4815 
 
 4867 
 
 4920 
 
 4972 
 
 5024 
 
 5076 
 
 5128 
 
 5180 
 
 52 
 
 1.92 
 
 0.6 5233 
 
 5285 
 
 5337 
 
 5389 
 
 5441 
 
 5493 
 
 5545 
 
 5596 
 
 5648 
 
 5700 
 
 52 
 
 1.93 
 
 0.6 5752 
 
 5804 
 
 5856 
 
 5907 
 
 5959 
 
 6011 
 
 6062 
 
 6114 
 
 6166 
 
 6217 
 
 52 
 
 1.94 
 
 0.6 6269 
 
 6320 
 
 6372 
 
 6423 
 
 6475 
 
 6526 
 
 6578 
 
 6629 
 
 6680 
 
 6732 
 
 52-51 
 
 1.95 
 
 0.6 6783 
 
 6834 
 
 6885 
 
 6937 
 
 6988 
 
 7039 
 
 7090 
 
 7141 
 
 7192 
 
 7243 
 
 51 
 
 1.96 
 
 0.6 7294 
 
 7345 
 
 7396 
 
 7447 
 
 7498 
 
 7549 
 
 7600 
 
 7651 
 
 7702 
 
 7753 
 
 51 
 
 1.97 
 
 0.6 7803 
 
 7854 
 
 7905 
 
 7956 
 
 8006 
 
 8057 
 
 8107 
 
 8158 
 
 8209 
 
 8259 
 
 51 
 
 1.98 
 
 0.6 8310 
 
 8360 
 
 8411 
 
 8461 
 
 8512 
 
 8562 
 
 8612 
 
 8663 
 
 8713 
 
 8763 
 
 50 
 
 1.99 
 
 0.6 8813 
 
 8864 
 
 8914 
 
 8964 
 
 9014 
 
 9064 
 
 9115 
 
 9165 
 
 9215 
 
 9265 
 
 50 
 
 2.00 
 
 0.6 9315 
 
 9365 
 
 9415 
 
 9465 
 
 9515 
 
 9564 
 
 9614 
 
 9664 
 
 9714 
 
 9764 
 
 50 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
132 
 
 TABLES. 
 Five-Place Natural Logarithms. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 2.00 
 
 0.6 9315 
 
 9365 
 
 9415 
 
 9465 
 
 9515 
 
 9564 
 
 9614 
 
 9664 
 
 9714 
 
 9764 
 
 50 
 
 2.01 
 
 0.6 9813 
 
 9863 
 
 9913 
 
 9963 - 
 
 *^0012 
 
 0062 
 
 0112 
 
 0161 
 
 0211 
 
 0260 
 
 50 
 
 2.02 
 
 0.7 0310 
 
 0359 
 
 0409 
 
 0458 
 
 0508 
 
 0557 
 
 0606 
 
 0656 
 
 0705 
 
 0754 
 
 49 
 
 2.03 
 
 0.7 0804 
 
 0853 
 
 0902 
 
 0951 
 
 1000 
 
 1050 
 
 1099 
 
 1148 
 
 1197 
 
 1246 
 
 49 
 
 2.04 
 
 0.7 1295 
 
 1344 
 
 1393 
 
 1442 
 
 1491 
 
 1540 
 
 1589 
 
 1638 
 
 1686 
 
 1735 
 
 49 
 
 2.05 
 
 0.7 1784 
 
 1833 
 
 1881 
 
 1930 
 
 1979 
 
 2028 
 
 2076 
 
 2125 
 
 2173 
 
 2222 
 
 49 
 
 2.06 
 
 0.7 2271 
 
 2319 
 
 2368 
 
 2416 
 
 2465 
 
 2513 
 
 2561 
 
 2610 
 
 2658 
 
 2707 
 
 49-48 
 
 2.07 
 
 0.7 2755 
 
 2803 
 
 2851 
 
 2900 
 
 2948 
 
 2996 
 
 3044 
 
 3092 
 
 3141 
 
 3189 
 
 48 
 
 2.08 
 
 0.7 3237 
 
 3285 
 
 3333 
 
 3381 
 
 3429 
 
 3477 
 
 3525 
 
 3573 
 
 3621 
 
 3669 
 
 48 
 
 2.09 
 
 0.7 3716 
 
 3764 
 
 3812 
 
 3860 
 
 3908 
 
 3955 
 
 4003 
 
 4051 
 
 4098 
 
 4146 
 
 48 
 
 2.10 
 
 0.7 4194 
 
 4241 
 
 4289 
 
 4336 
 
 4384 
 
 4432 
 
 4479 
 
 4527 
 
 4574 
 
 4621 
 
 48-47 
 
 2.11 
 
 0.7 4669 
 
 4716 
 
 4764 
 
 4811 
 
 4858 
 
 4905 
 
 4953 
 
 5000 
 
 5047 
 
 5094 
 
 47 
 
 2.12 
 
 0.7 5142 
 
 5189 
 
 5236 
 
 5283 
 
 5330 
 
 5377 
 
 5424 
 
 5471 
 
 5518 
 
 5565 
 
 47 
 
 2.13 
 
 0.7 5612 
 
 5659 
 
 5706 
 
 5753 
 
 5800 
 
 5847 
 
 5893 
 
 5940 
 
 5987 
 
 6034 
 
 47 
 
 2.14 
 
 0.7 6081 
 
 6127 
 
 6174 
 
 6221 
 
 6267 
 
 6314 
 
 6361 
 
 6407 
 
 6454 
 
 6500 
 
 47 
 
 2.LS 
 
 0.7 6547 
 
 6593 
 
 6640 
 
 6686 
 
 6733 
 
 6779 
 
 6825 
 
 6872 
 
 6918 
 
 6965 
 
 47-46 
 
 2.16 
 
 0.7 7011 
 
 7057 
 
 7103 
 
 7150 
 
 7196 
 
 7242 
 
 7288 
 
 7334 
 
 7381 
 
 7427 
 
 46 
 
 2.17 
 
 0.7 7473 
 
 7519 
 
 7565 
 
 7611 
 
 7657 
 
 7703 
 
 7749 
 
 7795 
 
 7841 
 
 7887 
 
 46 
 
 2.18 
 
 0.7 7932 
 
 7978 
 
 8024 
 
 8070 
 
 8116 
 
 8162 
 
 8207 
 
 8253 
 
 8299 
 
 8344 
 
 46 
 
 2.19 
 
 0.7 8390 
 
 8436 
 
 8481 
 
 8527 
 
 8573 
 
 8618 
 
 8664 
 
 8709 
 
 8755 
 
 8800 
 
 46-45 
 
 2.20 
 
 0.7 8846 
 
 8891 
 
 8937 
 
 8982 
 
 9027 
 
 9073 
 
 9118 
 
 9163 
 
 9209 
 
 9254 
 
 45 
 
 2.21 
 
 0.7 9299 
 
 9344 
 
 9390 
 
 9435 
 
 9480 
 
 9525 
 
 9570 
 
 9615 
 
 9661 
 
 9706 
 
 45 
 
 2.22 
 
 0.7 9751 
 
 9796 
 
 9841 
 
 9886 
 
 9931 
 
 9976 *0021 
 
 0066 
 
 0110 
 
 0155 
 
 45 
 
 2.23 
 
 0.8 0200 
 
 0245 
 
 0290 
 
 0335 
 
 0379 
 
 0424 
 
 0469 
 
 0514 
 
 0558 
 
 0603 
 
 45 
 
 2.24 
 
 0.8 0648 
 
 0692 
 
 0737 
 
 0781 
 
 0826 
 
 0871 
 
 0915 
 
 0960 
 
 1004 
 
 1049 
 
 45-44 
 
 2.25 
 
 0.8 1093 
 
 1137 
 
 1182 
 
 1226 
 
 1271 
 
 1315 
 
 1359 
 
 1404 
 
 1448 
 
 1492 
 
 44 
 
 2.26 
 
 0.8 1536 
 
 1581 
 
 1625 
 
 1669 
 
 1713 
 
 1757 
 
 1802 
 
 1846 
 
 1890 
 
 1934 
 
 44 
 
 2.27 
 
 0.8 1978 
 
 2022 
 
 2066 
 
 2110 
 
 2154 
 
 2198 
 
 2242 
 
 2286 
 
 2330 
 
 2374 
 
 44 
 
 2.28 
 
 0.8 2418 
 
 2461 
 
 2505 
 
 2549 
 
 2593 
 
 2637 
 
 2680 
 
 2724 
 
 2768 
 
 2812 
 
 44 
 
 2.29 
 
 0.8 2855 
 
 2899 
 
 2942 
 
 2986 
 
 3030 
 
 3073 
 
 3117 
 
 3160 
 
 3204 
 
 3247 
 
 44-43 
 
 2.30 
 
 0.8 3291 
 
 3334 
 
 3378 
 
 3421 
 
 3465 
 
 3508 
 
 3551 
 
 3595 
 
 3638 
 
 3681 
 
 43 
 
 2.31 
 
 0.8 3725 
 
 3768 
 
 3811 
 
 3855 
 
 3898 
 
 3941 
 
 3984 
 
 4027 
 
 4070 
 
 4114 
 
 43 
 
 2.32 
 
 0.8 4157 
 
 4200 
 
 4243 
 
 4286 
 
 4329 
 
 4372 
 
 4415 
 
 4458 
 
 4501 
 
 4544 
 
 43 
 
 2.33 
 
 0.8 4587 
 
 4630 
 
 4673 
 
 4715 
 
 4758 
 
 4801 
 
 4844 
 
 4887 
 
 4930 
 
 4972 
 
 43 
 
 2.34 
 
 0.8 5015 
 
 5058 
 
 5101 
 
 5143 
 
 5186 
 
 5229 
 
 5271 
 
 5314 
 
 5356 
 
 5399 
 
 43 
 
 2.35 
 
 0.8 5442 
 
 5484 
 
 5527 
 
 5569 
 
 5612 
 
 5654 
 
 5697 
 
 5739 
 
 5781 
 
 5824 
 
 43-48 
 
 2.36 
 
 0.8 5866 
 
 5909 
 
 5951 
 
 5993 
 
 6036 
 
 6078 
 
 6120 
 
 6162 
 
 6205 
 
 6247 
 
 42 
 
 2.37 
 
 0.8 6289 
 
 6331 
 
 6373 
 
 6415 
 
 6458 
 
 6500 
 
 6542 
 
 6584 
 
 6626 
 
 6668 
 
 42 
 
 2.38 
 
 0.8 6710 
 
 6752 
 
 6794 
 
 6836 
 
 6878 
 
 6920 
 
 6962 
 
 7004 
 
 7046 
 
 7087 
 
 42 
 
 2.39 
 
 0.8 7129 
 
 7171 
 
 7213 
 
 7255 
 
 7297 
 
 7338 
 
 7380 
 
 7422 
 
 7464 
 
 7505 
 
 42 
 
 2.40 
 
 0.8 7547 
 
 7589 
 
 7630 
 
 7672 
 
 7713 
 
 7755 
 
 7797 
 
 7838 
 
 7880 
 
 7921 
 
 42 
 
 2.41 
 
 0.8 7963 
 
 8004 
 
 8046 
 
 8087 
 
 8129 
 
 8170 
 
 8211 
 
 8253 
 
 8294 
 
 8335 
 
 41 
 
 2.42 
 
 0.8 8377 
 
 8418 
 
 8459 
 
 8501 
 
 8542 
 
 8583 
 
 8624 
 
 8666 
 
 8707 
 
 8748 
 
 41 
 
 2.43 
 
 0.8 8789 
 
 8830 
 
 8871 
 
 8913 
 
 8954 
 
 8995 
 
 9036 
 
 9077 
 
 9118 
 
 9159 
 
 41 
 
 2.44 
 
 0.8 9200 
 
 9241 
 
 9282 
 
 9323 
 
 9364 
 
 9405 
 
 9445 
 
 9486 
 
 9527 
 
 9568 
 
 41 
 
 2.45 
 
 0.8 9609 
 
 9650 
 
 9690 
 
 9731 
 
 9772 
 
 9813 
 
 9853 
 
 9894 
 
 9935 
 
 9975 
 
 41 
 
 2.46 
 
 0.9 0016 
 
 0057 
 
 0097 
 
 0138 
 
 0179 
 
 0219 
 
 0260 
 
 0300 
 
 0341 
 
 0381 
 
 41-40 
 
 2.47 
 
 0.9 0422 
 
 0462 
 
 0503 
 
 0543 
 
 0584 
 
 0624 
 
 0664 
 
 0705 
 
 0745 
 
 0786 
 
 40 
 
 2.48 
 
 0.9 0826 
 
 0866 
 
 0906 
 
 0947 
 
 0987 
 
 1027 
 
 1067 
 
 1108 
 
 1148 
 
 1188 
 
 40 
 
 2.49 
 
 0.9 1228 
 
 1268 
 
 1309 
 
 1349 
 
 1389 
 
 1429 
 
 1469 
 
 1509 
 
 1549 
 
 1589 
 
 40 
 
 2.60 
 
 0.91629 
 
 1669 
 
 1709 
 
 1749 
 
 1789 
 
 1829 
 
 1869 
 
 1909 
 
 1949 
 
 1988 
 
 40 
 
 
 
 
 J 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 I 
 
TABLES. 
 
 133 
 
 Five-Place Natural Logarithms. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 2.50 
 
 0.91629 
 
 1669 
 
 1709 
 
 1749 
 
 1789 
 
 1829 
 
 1869 
 
 1909 
 
 1949 
 
 1988 
 
 40 
 
 2.51 
 
 0.9 2028 
 
 2068 
 
 2108 
 
 2148 
 
 2188 
 
 2227 
 
 2267 
 
 2307 
 
 2346 
 
 2386 
 
 40 
 
 2.52 
 
 0.9 2426 
 
 2466 
 
 2505 
 
 2545 
 
 2584 
 
 2624 
 
 2664 
 
 2703 
 
 2743 
 
 2782 
 
 40 
 
 2.53 
 
 0.9 2S22 
 
 2S61 
 
 2901 
 
 2940 
 
 2980 
 
 3019 
 
 3059 
 
 3098 
 
 3138 
 
 3177 
 
 40-39 
 
 2.54 
 
 0.9 3216 
 
 3256 
 
 3295 
 
 3334 
 
 3374 
 
 3413 
 
 3452 
 
 3492 
 
 3531 
 
 3570 
 
 39 
 
 2.55 
 
 0.9 3609 
 
 3649 
 
 3688 
 
 3727 
 
 3766 
 
 3805 
 
 3844 
 
 3883 
 
 3923 
 
 3962 
 
 39 
 
 2.56 
 
 0.9 4001 
 
 4040 
 
 4079 
 
 4118 
 
 4157 
 
 4196 
 
 4235 
 
 4274 
 
 4313 
 
 4352 
 
 39 
 
 2.57 
 
 0.9 4391 
 
 4429 
 
 4468 
 
 4507 
 
 4546 
 
 4585 
 
 4624 
 
 4663 
 
 4701 
 
 4740 
 
 39 
 
 2.58 
 
 0.9 4779 
 
 4818 
 
 4856 
 
 4895 
 
 4934 
 
 4973 
 
 5011 
 
 5050 
 
 5089 
 
 5127 
 
 39 
 
 2.59 
 
 0.9 5166 
 
 5204 
 
 5243 
 
 5282 
 
 5320 
 
 5359 
 
 5397 
 
 5436 
 
 5474 
 
 5513 
 
 39-38 
 
 2.60 
 
 0.9 5551 
 
 5590 
 
 5628 
 
 5666 
 
 5705 
 
 5743 
 
 5782 
 
 5820 
 
 5858 
 
 5897 
 
 38 
 
 2.61 
 
 0.9 5935 
 
 5973 
 
 6012 
 
 6050 
 
 6088 
 
 6126 
 
 6165 
 
 6203 
 
 6241 
 
 6279 
 
 38 
 
 2.62 
 
 0.9 6317 
 
 6356 
 
 6394 
 
 6432 
 
 6470 
 
 6508 
 
 6546 
 
 6584 
 
 6622 
 
 6660 
 
 38 
 
 2.63 
 
 0.9 6698 
 
 6736 
 
 6774 
 
 6812 
 
 6S50 
 
 6S88 
 
 6926 
 
 6964 
 
 7002 
 
 7040 
 
 38 
 
 2.64 
 
 0.9 7078 
 
 7116 
 
 7154 
 
 7191 
 
 7229 
 
 7267 
 
 7305 
 
 7343 
 
 7380 
 
 7418 
 
 38 
 
 2.65 
 
 0.9 7456 
 
 7494 
 
 7531 
 
 7569 
 
 7607 
 
 7644 
 
 7682 
 
 7720 
 
 7757 
 
 7795 
 
 38 
 
 2.66 
 
 0.9 7833 
 
 7S70 
 
 7908 
 
 7945 
 
 7983 
 
 8020 
 
 8058 
 
 8095 
 
 8133 
 
 8170 
 
 38-37 
 
 2.67 
 
 0.9 8208 
 
 8245 
 
 8283 
 
 8320 
 
 8358 
 
 8395 
 
 8432 
 
 8470 
 
 8507 
 
 8544 
 
 37 
 
 2.68 
 
 0.9 8582 
 
 8619 
 
 8656 
 
 8694 
 
 8731 
 
 8768 
 
 8805 
 
 8843 
 
 8880 
 
 8917 
 
 37 
 
 2.69 
 
 0.9 8954 
 
 8991 
 
 9028 
 
 9066 
 
 9103 
 
 9140 
 
 9177 
 
 9214 
 
 9251 
 
 9288 
 
 37 
 
 2.70 
 
 0.9 9325 
 
 9362 
 
 9399 
 
 9436 
 
 9473 
 
 9510 
 
 9547 
 
 9584 
 
 9621 
 
 9658 
 
 37 
 
 2.71 
 
 0.9 9695 
 
 9732 
 
 9769 
 
 9806 
 
 9842 
 
 9879 
 
 9916 
 
 9953 
 
 9990 
 
 *0026 
 
 37 
 
 2.72 
 
 1.0 0063 
 
 0100 
 
 0137 
 
 0173 
 
 0210 
 
 0247 
 
 0284 
 
 0320 
 
 0357 
 
 0394 
 
 37 
 
 2.73 
 
 1.0 0430 
 
 0467 
 
 0503 
 
 0540 
 
 0577 
 
 0613 
 
 0650 
 
 0686 
 
 0723 
 
 0759 
 
 37 
 
 2.74 
 
 1.0 0796 
 
 0832 
 
 0869 
 
 0905 
 
 0942 
 
 0978 
 
 1015 
 
 1051 
 
 1087 
 
 1124 
 
 36 
 
 2.75 
 
 1.0 1160 
 
 1196 
 
 1233 
 
 1269 
 
 1305 
 
 1342 
 
 1378 
 
 1414 
 
 1451 
 
 1487 
 
 36 
 
 2.76 
 
 1.0 1523 
 
 1559 
 
 1596 
 
 1632 
 
 1668 
 
 1704 
 
 1740 
 
 1776 
 
 1813 
 
 1849 
 
 36 
 
 2.77 
 
 1.0 1S85 
 
 1921 
 
 1957 
 
 1993 
 
 2029 
 
 2065 
 
 2101 
 
 2137 
 
 2173 
 
 2209 
 
 36 
 
 2.78 
 
 1.0 2245 
 
 2281 
 
 2317 
 
 2353 
 
 2389 
 
 2425 
 
 2461 
 
 2497 
 
 2532 
 
 2588 
 
 36 
 
 2.79 
 
 1.0 2604 
 
 2640 
 
 2676 
 
 2712 
 
 2747 
 
 2783 
 
 2819 
 
 2855 
 
 2890 
 
 2926 
 
 36 
 
 2.80 
 
 1.0 2962 
 
 2998 
 
 3033 
 
 3069 
 
 3105 
 
 3140 
 
 3176 
 
 3212 
 
 3247 
 
 3283 
 
 36 
 
 2.81 
 
 1.0 3318 
 
 3354 
 
 3390 
 
 3425 
 
 3461 
 
 3496 
 
 3532 
 
 3567 
 
 3603 
 
 3638 
 
 36-35 
 
 2.82 
 
 1.0 3674 
 
 3709 
 
 3745 
 
 3780 
 
 3815 
 
 3851 
 
 3886 
 
 3922 
 
 3957 
 
 3992 
 
 35 
 
 2.S3 
 
 1.0 4028 
 
 4063 
 
 4098 
 
 4134 
 
 4169 
 
 4204 
 
 4239 
 
 4275 
 
 4310 
 
 4345 
 
 35 
 
 2.84 
 
 1.0 43 SO 
 
 4416 
 
 4451 
 
 4486 
 
 4521 
 
 4556 
 
 4591 
 
 4627 
 
 4662 
 
 4697 
 
 35 
 
 2.85 
 
 1.0 4732 
 
 4767 
 
 4802 
 
 4837 
 
 4872 
 
 4907 
 
 4942 
 
 4977 
 
 5012 
 
 5047 
 
 35 
 
 2.86 
 
 1.0 5082 
 
 5117 
 
 5152 
 
 5187 
 
 5222 
 
 5257 
 
 5292 
 
 5327 
 
 5361 
 
 5396 
 
 35 
 
 2.87 
 
 1.0 5431 
 
 5466 
 
 5501 
 
 5536 
 
 5570 
 
 5605 
 
 5640 
 
 5675 
 
 5710 
 
 5744 
 
 35 
 
 2.88 
 
 1.0 5779 
 
 5814 
 
 5848 
 
 5883 
 
 5918 
 
 5952 
 
 5987 
 
 6022 
 
 6056 
 
 6091 
 
 35 
 
 2.89 
 
 1.0 6126 
 
 6160 
 
 6195 
 
 6229 
 
 6264 
 
 6299 
 
 6333 
 
 6368 
 
 6402 
 
 6437 
 
 35-34 
 
 2.90 
 
 1.0 6471 
 
 6506 
 
 6540 
 
 6574 
 
 6609 
 
 6643 
 
 6678 
 
 6712 
 
 6747 
 
 6781 
 
 34 
 
 2.91 
 
 1.0 6815 
 
 6850 
 
 6884 
 
 6918 
 
 6953 
 
 6987 
 
 7021 
 
 7056 
 
 7090 
 
 7124 
 
 34 
 
 2.92 
 
 1.0 7158 
 
 7193 
 
 7227 
 
 7261 
 
 7295 
 
 7329 
 
 7364 
 
 7398 
 
 7432 
 
 7466 
 
 34 
 
 2.93 
 
 1.0 7500 
 
 7534 
 
 7568 
 
 7603 
 
 7637 
 
 7671 
 
 7705 
 
 7739 
 
 7773 
 
 7807 
 
 34 
 
 2.94 
 
 1.0 7841 
 
 7875 
 
 7909 
 
 7943 
 
 7977 
 
 8011 
 
 8045 
 
 8079 
 
 8113 
 
 8147 
 
 34 
 
 2.95 
 
 1.0 8181 
 
 8214 
 
 8248 
 
 8282 
 
 8316 
 
 8350 
 
 8384 
 
 8418 
 
 8451 
 
 8485 
 
 34 
 
 2.96 
 
 1.0 8519 
 
 8553 
 
 8586 
 
 8620 
 
 8654 
 
 8688 
 
 8721 
 
 8755 
 
 8789 
 
 8823 
 
 34 
 
 2.97 
 
 1.0 8856 
 
 8890 
 
 8924 
 
 8957 
 
 8991 
 
 9024 
 
 9058 
 
 9092 
 
 9125 
 
 9159 
 
 34 
 
 2.98 
 
 1.0 9192 
 
 9226 
 
 9259 
 
 9293 
 
 9326 
 
 9360 
 
 9393 
 
 9427 
 
 9460 
 
 9494 
 
 34-33 
 
 2.99 
 
 1.0 9527 
 
 9561 
 
 9594 
 
 9628 
 
 9661 
 
 9694 
 
 9728 
 
 9761 
 
 9795 
 
 9828 
 
 33 
 
 3.00 
 
 1.0 9861 
 
 9895 
 
 9928 
 
 9961 
 
 9994 
 
 *0028 
 
 0061 
 
 0094 
 
 0128 
 
 0161 
 
 33 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
134 
 
 TABLES. 
 
 Five-Place Natural Logarithms. 
 
 No, 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 3.00 
 
 1.0 9861 
 
 9895 
 
 9928 
 
 9961 
 
 9994 
 
 *0028 
 
 0061 
 
 0094 
 
 0128 
 
 0161 
 
 33 
 
 3.01 
 
 1 10194 
 
 0227 
 
 0260 
 
 0294 
 
 0327 
 
 0360 
 
 0393 
 
 0426 
 
 0459 
 
 0493 
 
 33 
 
 3.02 
 
 i.l 0526 
 
 0559 
 
 0592 
 
 0625 
 
 0658 
 
 0691 
 
 0724 
 
 0757 
 
 0790 
 
 0823 
 
 33 
 
 3.03 
 
 1.1 0856 
 
 0889 
 
 0922 
 
 0955 
 
 0988 
 
 1021 
 
 1054 
 
 1087 
 
 1120 
 
 1153 
 
 33 
 
 3.04 
 
 1.1 1186 
 
 1219 
 
 1252 
 
 1284 
 
 1317 
 
 1350 
 
 1383 
 
 1416 
 
 1449 
 
 1481 
 
 33 
 
 3.05 
 
 1.11514 
 
 1547 
 
 1580 
 
 1612 
 
 1645 
 
 1678 
 
 1711 
 
 1743 
 
 1776 
 
 1809 
 
 33 
 
 3.06 
 
 1.1 1841 
 
 1874 
 
 1907 
 
 1939 
 
 1972 
 
 2005 
 
 2037 
 
 2070 
 
 2103 
 
 2135 
 
 33 
 
 3.07 
 
 1.1 2168 
 
 2200 
 
 2233 
 
 2265 
 
 2298 
 
 2330 
 
 2363 
 
 2396 
 
 2428 
 
 2460 
 
 33-33 
 
 3.08 
 
 1.1 2493 
 
 2525 
 
 2558 
 
 2590 
 
 2623 
 
 2655 
 
 2688 
 
 2720 
 
 2752 
 
 2785 
 
 32 
 
 3.09 
 
 1.12817 
 
 2849 
 
 2882 
 
 2914 
 
 2946 
 
 2979 
 
 3011 
 
 3043 
 
 3076 
 
 3108 
 
 32 
 
 8.10 
 
 1.13140 
 
 3172 
 
 3205 
 
 3237 
 
 3269 
 
 3301 
 
 3334 
 
 3366 
 
 3398 
 
 3430 
 
 32 
 
 3.11 
 
 1.1 3462 
 
 3494 
 
 3527 
 
 3559 
 
 3591 
 
 3623 
 
 3655 
 
 3687 
 
 3719 
 
 3751 
 
 32 
 
 3.12 
 
 1.1 3783 
 
 3815 
 
 3847 
 
 3879 
 
 3911 
 
 3943 
 
 3955 
 
 4007 
 
 4039 
 
 4071 
 
 32 
 
 3.13 
 
 1.1 4103 
 
 4135 
 
 4167 
 
 4199 
 
 4231 
 
 4263 
 
 4295 
 
 4327 
 
 4359 
 
 4390 
 
 32 
 
 3.14 
 
 1.14422 
 
 4454 
 
 4486 
 
 4518 
 
 4550 
 
 4581 
 
 4613 
 
 4645 
 
 4677 
 
 4708 
 
 32 
 
 3.15 
 
 1.1 4740 
 
 4772 
 
 4804 
 
 4835 
 
 4867 
 
 4899 
 
 4931 
 
 4962 
 
 4994 
 
 5026 
 
 32 
 
 3.16 
 
 1.1 5057 
 
 5089 
 
 5120 
 
 5152 
 
 5184 
 
 5215 
 
 5247 
 
 5278 
 
 5310 
 
 5342 
 
 32 
 
 3.17 
 
 1.15373 
 
 5405 
 
 5436 
 
 5468 
 
 5499 
 
 5531 
 
 5562 
 
 5594 
 
 5625 
 
 5657 
 
 32-31 
 
 3.18 
 
 1.1 5688 
 
 5720 
 
 5751 
 
 5782 
 
 5814 
 
 5845 
 
 5877 
 
 5908 
 
 5939 
 
 5971 
 
 31 
 
 3.19 
 
 1.1 6002 
 
 6033 
 
 6065 
 
 6096 
 
 6127 
 
 6159 
 
 6190 
 
 6221 
 
 6253 
 
 6284 
 
 31 
 
 3.20 
 
 1.1 6315 
 
 6346 
 
 6378 
 
 6409 
 
 6440 
 
 6471 
 
 6502 
 
 6534 
 
 6565 
 
 6596 
 
 31 
 
 3.21 
 
 1.1 6627 
 
 6658 
 
 6689 
 
 6721 
 
 6752 
 
 6783 
 
 6814 
 
 6845 
 
 6876 
 
 6907 
 
 31 
 
 3.22 
 
 1.1 6938 
 
 6969 
 
 7000 
 
 7031 
 
 7062 
 
 7093 
 
 7124 
 
 7155 
 
 7186 
 
 7217 
 
 31 
 
 3.23 
 
 1.1 7248 
 
 7279 
 
 7310 
 
 7341 
 
 7372 
 
 7403 
 
 7434 
 
 7465 
 
 7496 
 
 7526 
 
 31 
 
 3.24 
 
 1.1 7557 
 
 7588 
 
 7619 
 
 7650 
 
 7681 
 
 7712 
 
 7742 
 
 7773 
 
 7804 
 
 7835 
 
 31 
 
 3.25 
 
 1.1 7865 
 
 7896 
 
 7927 
 
 7958 
 
 7989 
 
 8019 
 
 8050 
 
 8081 
 
 8111 
 
 8142 
 
 31 
 
 3.26 
 
 1.18173 
 
 8203 
 
 8234 
 
 8265 
 
 8295 
 
 8326 
 
 8357 
 
 8387 
 
 8418 
 
 8448 
 
 31 
 
 3.27 
 
 1.1 8479 
 
 8510 
 
 8540 
 
 8571 
 
 8601 
 
 8632 
 
 8662 
 
 8693 
 
 8723 
 
 8754 
 
 31-30 
 
 3.28 
 
 1.1 8784 
 
 8815 
 
 8845 
 
 8876 
 
 8906 
 
 8937 
 
 8967 
 
 8998 
 
 9028 
 
 9058 
 
 30 
 
 3.29 
 
 1.1 9089 
 
 9119 
 
 9150 
 
 9180 
 
 9210 
 
 9241 
 
 9271 
 
 9301 
 
 9332 
 
 9362 
 
 30 
 
 3.30 
 
 1.1 9392 
 
 9423 
 
 9453 
 
 9483 
 
 9513 
 
 9544 
 
 9574 
 
 9604 
 
 9634 
 
 9665 
 
 30 
 
 3.31 
 
 1.1 9695 
 
 9725 
 
 9755 
 
 9785 
 
 9816 
 
 9846 
 
 9876 
 
 9906 
 
 9936 
 
 9966 
 
 30 
 
 3.32 
 
 1.1 9996 
 
 *0027 
 
 0057 
 
 0087 
 
 0117 
 
 0147 
 
 0177 
 
 0207 
 
 0237 
 
 0267 
 
 30 
 
 3.33 
 
 1.2 0297 
 
 0327 
 
 0357 
 
 0387 
 
 0417 
 
 0447 
 
 0477 
 
 0507 
 
 0537 
 
 0567 
 
 30 
 
 3.34 
 
 1.2 0597 
 
 0627 
 
 0657 
 
 06S7 
 
 0717 
 
 0747 
 
 0777 
 
 0806 
 
 0836 
 
 0866 
 
 30 
 
 3.35 
 
 1.2 0896 
 
 0926 
 
 0956 
 
 0986 
 
 1015 
 
 1045 
 
 1075 
 
 1105 
 
 1135 
 
 1164 
 
 30 
 
 3.36 
 
 1.21194 
 
 1224 
 
 1254 
 
 1283 
 
 1313 
 
 1343 
 
 1373 
 
 1402 
 
 1432 
 
 1462 
 
 30 
 
 3.37 
 
 1.2 1491 
 
 1521 
 
 1551 
 
 1580 
 
 1610 
 
 1640 
 
 1669 
 
 1699 
 
 1728 
 
 1758 
 
 30 
 
 3.38 
 
 1.2 1788 
 
 1817 
 
 1847 
 
 1876 
 
 1906 
 
 1935 
 
 1965 
 
 1994 
 
 2024 
 
 2053 
 
 30 
 
 3.39 
 
 1.2 2083 
 
 2112 
 
 2142 
 
 2171 
 
 2201 
 
 2230 
 
 2260 
 
 2289 
 
 2319 
 
 2348 
 
 29 
 
 3.40 
 
 1.2 2378 
 
 2407 
 
 2436 
 
 2466 
 
 2495 
 
 2524 
 
 2554 
 
 2583 
 
 2613 
 
 2642 
 
 29 
 
 3.41 
 
 1.2 2671 
 
 2701 
 
 2730 
 
 2759 
 
 2788 
 
 2818 
 
 2847 
 
 2876 
 
 2906 
 
 2935 
 
 29 
 
 3.42 
 
 1.2 2964 
 
 2993 
 
 3023 
 
 3052 
 
 3081 
 
 3110 
 
 3139 
 
 3169 
 
 3198 
 
 3227 
 
 29 
 
 3.43 
 
 1.2 3256 
 
 3285 
 
 3314 
 
 3343 
 
 3373 
 
 3402 
 
 3431 
 
 3460 
 
 3489 
 
 3518 
 
 29 
 
 3.44 
 
 1.2 3547 
 
 3576 
 
 3605 
 
 3634 
 
 3663 
 
 3692 
 
 3721 
 
 3750 
 
 3779 
 
 3808 
 
 29 
 
 3.45 
 
 1.2 3837 
 
 3866 
 
 3895 
 
 3924 
 
 3953 
 
 3982 
 
 4011 
 
 4040 
 
 4069 
 
 4098 
 
 29 
 
 3.46 
 
 1.2 4127 
 
 4156 
 
 4185 
 
 4214 
 
 4242 
 
 4271 
 
 4300 
 
 4329 
 
 4358 
 
 4387 
 
 29 
 
 3.47 
 
 1.2 4415 
 
 4444 
 
 4473 
 
 4502 
 
 4531 
 
 4559 
 
 4588 
 
 4617 
 
 4646 
 
 4674 
 
 29 
 
 3.48 
 
 1.2 4703 
 
 4732 
 
 4761 
 
 4789 
 
 4818 
 
 4847 
 
 4875 
 
 4904 
 
 4933 
 
 4962 
 
 29 
 
 3.49 
 
 1.2 4990 
 
 5019 
 
 5047 
 
 5076 
 
 5105 
 
 5133 
 
 5162 
 
 5191 
 
 5219 
 
 5248 
 
 29 
 
 8.50 
 
 1.2 5276 
 
 5305 
 
 5333 
 
 5362 
 
 5391 
 
 5419 
 
 5448 
 
 5476 
 
 5505 
 
 5533 
 
 29-28 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
TABLES. 
 Five-Place Natural Logarithms. 
 
 135 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 3.50 
 
 1.2 5276 
 
 5305 
 
 5333 
 
 5362 
 
 5391 
 
 5419 
 
 5448 
 
 5476 
 
 5505 
 
 5533 
 
 29-28 
 
 3.51 
 
 1.2 5562 
 
 5590 
 
 5619 
 
 5647 
 
 5675 
 
 5704 
 
 5732 
 
 5761 
 
 5789 
 
 5818 
 
 28 
 
 3.52 
 
 1.2 5846 
 
 5875 
 
 5903 
 
 5931 
 
 5960 
 
 5988 
 
 6016 
 
 6045 
 
 6073 
 
 6101 
 
 28 
 
 3.53 
 
 1.2 6130 
 
 6158 
 
 6186 
 
 6215 
 
 6243 
 
 6271 
 
 6300 
 
 6328 
 
 6356 
 
 6384 
 
 28 
 
 3.54 
 
 1.2 6413 
 
 6441 
 
 6469 
 
 6497 
 
 6526 
 
 6554 
 
 6582 
 
 6610 
 
 6638 
 
 6667 
 
 28 
 
 3.55 
 
 1.2 6695 
 
 6723 
 
 6751 
 
 6779 
 
 6807 
 
 6836 
 
 6864 
 
 6892 
 
 6920 
 
 6948 
 
 28 
 
 3.56 
 
 1.2 6976 
 
 7004 
 
 7032 
 
 7060 
 
 7088 
 
 7116 
 
 7144 
 
 7172 
 
 7201 
 
 7229 
 
 28 
 
 3.57 
 
 1.2 7257 
 
 7285 
 
 7313 
 
 7341 
 
 7369 
 
 7397 
 
 7424 
 
 7452 
 
 7480 
 
 7508 
 
 28 
 
 3.5S 
 
 1.2 7536 
 
 7564 
 
 7592 
 
 7620 
 
 7648 
 
 7676 
 
 7704 
 
 7732 
 
 7759 
 
 7787 
 
 28 
 
 3.59 
 
 1.2 7815 
 
 7843 
 
 7871 
 
 7899 
 
 7927 
 
 7954 
 
 7982 
 
 8010 
 
 8038 
 
 8066 
 
 28 
 
 3.60 
 
 1.2 8093 
 
 8121 
 
 8149 
 
 8177 
 
 8204 
 
 8232 
 
 8260 
 
 8288 
 
 8315 
 
 8343 
 
 28 
 
 3.61 
 
 1.2 8371 
 
 8398 
 
 8426 
 
 8454 
 
 8482 
 
 8509 
 
 8537 
 
 8564 
 
 8592 
 
 8620 
 
 28 
 
 3.62 
 
 1.2 8647 
 
 8675 
 
 8703 
 
 8730 
 
 8758 
 
 8785 
 
 8813 
 
 8841 
 
 8868 
 
 8896 
 
 28 
 
 3.63 
 
 1.2 8923 
 
 8951 
 
 8978 
 
 9006 
 
 9033 
 
 9061 
 
 9088 
 
 9116 
 
 9143 
 
 9171 
 
 28-27 
 
 3.64 
 
 1.2 9198 
 
 9226 
 
 9253 
 
 9281 
 
 9308 
 
 9336 
 
 9363 
 
 9390 
 
 9418 
 
 9445 
 
 27 
 
 3.65 
 
 1.2 9473 
 
 9500 
 
 9527 
 
 9555 
 
 9582 
 
 9610 
 
 9637 
 
 9664 
 
 9692 
 
 9719 
 
 27 
 
 3.66 
 
 1.2 9746 
 
 9774 
 
 9801 
 
 9828 
 
 9856 
 
 9883 
 
 9910 
 
 9937 
 
 9965 
 
 9992 
 
 27 
 
 3.67 
 
 1.3 0019 
 
 0046 
 
 0074 
 
 0101 
 
 0128 
 
 0155 
 
 0183 
 
 0210 
 
 0237 
 
 0264 
 
 27 
 
 3.68 
 
 1.3 0291 
 
 0318 
 
 0346 
 
 0373 
 
 0400 
 
 0427 
 
 0454 
 
 0481 
 
 0508 
 
 0536 
 
 27 
 
 3.69 
 
 1.3 0563 
 
 0590 
 
 0617 
 
 0644 
 
 0671 
 
 0698 
 
 0725 
 
 0752 
 
 0779 
 
 0806 
 
 27 
 
 3.70 
 
 1.3 0833 
 
 0860 
 
 0887 
 
 0914 
 
 0941 
 
 0968 
 
 0995 
 
 1022 
 
 1049 
 
 1076 
 
 27 
 
 3.71 
 
 1.3 1103 
 
 1130 
 
 1157 
 
 1184 
 
 1211 
 
 1238 
 
 1265 
 
 1292 
 
 1319 
 
 1345 
 
 27 
 
 3.72 
 
 1.3 1372 
 
 1399 
 
 1426 
 
 1453 
 
 1480 
 
 1507 
 
 1534 
 
 1560 
 
 1587 
 
 1614 
 
 27 
 
 3.73 
 
 1.3 1641 
 
 1668 
 
 1694 
 
 1721 
 
 1748 
 
 1775 
 
 1802 
 
 1828 
 
 1855 
 
 1882 
 
 27 
 
 3.74 
 
 1.3 1909 
 
 1935 
 
 1962 
 
 1989 
 
 2015 
 
 2042 
 
 2069 
 
 2096 
 
 2122 
 
 2149 
 
 27 
 
 3.75 
 
 1.3 2176 
 
 2202 
 
 2229 
 
 2256 
 
 2282 
 
 2309 
 
 2335 
 
 2362 
 
 2389 
 
 2415 
 
 27 
 
 3.76 
 
 1.3 2442 
 
 2468 
 
 2495 
 
 2522 
 
 2548 
 
 2575 
 
 2601 
 
 2628 
 
 2654 
 
 2681 
 
 27 
 
 3.77 
 
 1.3 2708 
 
 2734 
 
 2761 
 
 2787 
 
 2814 
 
 2840 
 
 2867 
 
 2893 
 
 2919 
 
 2946 
 
 27-26 
 
 3.78 
 
 1.3 2972 
 
 2999 
 
 3025 
 
 3052 
 
 3078 
 
 3105 
 
 3131 
 
 3157 
 
 3184 
 
 3210 
 
 26 
 
 3.79 
 
 1.3 3237 
 
 3263 
 
 3289 
 
 3316 
 
 3342 
 
 3368 
 
 3395 
 
 3421 
 
 3447 
 
 3474 
 
 26 
 
 3.80 
 
 1.3 3500 
 
 3526 
 
 3553 
 
 3579 
 
 3605 
 
 3632 
 
 3658 
 
 3684 
 
 3710 
 
 3737 
 
 26 
 
 3.81 
 
 1.3 3763 
 
 3789 
 
 3815 
 
 3842 
 
 3868 
 
 3894 
 
 3920 
 
 3946 
 
 3973 
 
 3999 
 
 26 
 
 3.82 
 
 1.3 4025 
 
 4051 
 
 4077 
 
 4104 
 
 4130 
 
 4156 
 
 4182 
 
 4208 
 
 4234 
 
 4260 
 
 26 
 
 3.83 
 
 1.3 4286 
 
 4313 
 
 4339 
 
 4365 
 
 4391 
 
 4417 
 
 4443 
 
 4469 
 
 4495 
 
 4521 
 
 26 
 
 3.84 
 
 1.3 4547 
 
 4573 
 
 4599 
 
 4625 
 
 4651 
 
 4677 
 
 4703 
 
 4729 
 
 4755 
 
 4781 
 
 26 
 
 3.85 
 
 1.3 4807 
 
 4833 
 
 4859 
 
 4885 
 
 4911 
 
 4937 
 
 4963 
 
 4989 
 
 5015 
 
 5041 
 
 26 
 
 3.86 
 
 1.3 5067 
 
 5093 
 
 5119 
 
 5144 
 
 5170 
 
 5196 
 
 5222 
 
 5248 
 
 5274 
 
 5300 
 
 26 
 
 3.87 
 
 1.3 5325 
 
 5351 
 
 5377 
 
 5403 
 
 5429 
 
 5455 
 
 5480 
 
 5506 
 
 5532 
 
 5558 
 
 26 
 
 3.88 
 
 1.3 5584 
 
 5609 
 
 5635 
 
 5661 
 
 5687 
 
 5712 
 
 5738 
 
 5764 
 
 5789 
 
 5815 
 
 26 
 
 3.89 
 
 1.3 5841 
 
 5867 
 
 5892 
 
 5918 
 
 5944 
 
 5969 
 
 5995 
 
 6021 
 
 6046 
 
 6072 
 
 26 
 
 3.90 
 
 1.3 6098 
 
 6123 
 
 6149 
 
 6175 
 
 6200 
 
 6226 
 
 6251 
 
 6277 
 
 6303 
 
 6328 
 
 26 
 
 3.91 
 
 1.3 6354 
 
 6379 
 
 6405 
 
 6430 
 
 6456 
 
 6481 
 
 6507 
 
 6533 
 
 6558 
 
 6584 
 
 26 
 
 3.92 
 
 1.3 6609 
 
 6635 
 
 6660 
 
 6686 
 
 6711 
 
 6737 
 
 6762 
 
 6788 
 
 6813 
 
 6838 
 
 26-25 
 
 3.93 
 
 1.3 6864 
 
 6889 
 
 6915 
 
 6940 
 
 6966 
 
 6991 
 
 7016 
 
 7042 
 
 7067 
 
 7093 
 
 25 
 
 3.94 
 
 1.3 7118 
 
 7143 
 
 7169 
 
 7194 
 
 7220 
 
 7245 
 
 7270 
 
 7296 
 
 7321 
 
 7346 
 
 25 
 
 3.95 
 
 1.3 7372 
 
 7397 
 
 7422 
 
 7447 
 
 7473 
 
 7498 
 
 7523 
 
 7549 
 
 7574 
 
 7599 
 
 25 
 
 3.96 
 
 1.3 7624 
 
 7650 
 
 7675 
 
 7700 
 
 7725 
 
 7751 
 
 7776 
 
 7801 
 
 7826 
 
 7851 
 
 25 
 
 3.97 
 
 1.3 7877 
 
 7902 
 
 7927 
 
 7952 
 
 7977 
 
 8002 
 
 8028 
 
 8053 
 
 8078 
 
 8103 
 
 25 
 
 3.98 
 
 1.3 8128 
 
 81-13 
 
 8178 
 
 8204 
 
 8229 
 
 8254 
 
 8279 
 
 8304 
 
 8329 
 
 8354 
 
 25 
 
 3.99 
 
 1.3 8379 
 
 8404 
 
 8429 
 
 8454 
 
 8479 
 
 8504 
 
 8529 
 
 8554 
 
 8579 
 
 8604 
 
 25 
 
 4.00 
 
 1.3 8629 
 
 8654 
 
 8679 
 
 8704 
 
 8729 
 
 8754 
 
 8779 
 
 8804 
 
 8829 
 
 8854 
 
 25 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
136 
 
 TABLES. 
 Five-Place Natural Logarithms. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 4.00 
 
 1.3 8629 
 
 8654 
 
 8679 
 
 8704 
 
 8729 
 
 8754 
 
 8779 
 
 8804 
 
 8829 
 
 8854 
 
 25 
 
 4.01 
 
 1.3 8879 
 
 8904 
 
 8929 
 
 8954 
 
 8979 
 
 9004 
 
 9029 
 
 9054 
 
 9078 
 
 9103 
 
 25 
 
 4.02 
 
 1.3 9128 
 
 9153 
 
 9178 
 
 9203 
 
 9228 
 
 9252 
 
 9277 
 
 9302 
 
 9327 
 
 9352 
 
 25 
 
 4.03 
 
 1.3 9377 
 
 9401 
 
 9426 
 
 9451 
 
 9476 
 
 9501 
 
 9525 
 
 9550 
 
 9575 
 
 9600 
 
 25 
 
 4.04 
 
 1.3 9624 
 
 9649 
 
 9674 
 
 9699 
 
 9723 
 
 9748 
 
 9773 
 
 9798 
 
 9822 
 
 9847 
 
 25 
 
 4.0.S 
 
 1.3 9872 
 
 9896 
 
 9921 
 
 9946 
 
 9970 
 
 9995 
 
 *0020 
 
 0044 
 
 0069 
 
 0094 
 
 25 
 
 4.06 
 
 1.4 0118 
 
 0143 
 
 0168 
 
 0192 
 
 0217 
 
 0241 
 
 0266 
 
 0291 
 
 0315 
 
 0340 
 
 25 
 
 4.07 
 
 1.4 0364 
 
 0389 
 
 0413 
 
 0438 
 
 04b3 
 
 0487 
 
 0512 
 
 0536 
 
 0561 
 
 0585 
 
 25 
 
 4.08 
 
 1.4 0610 
 
 0634 
 
 0659 
 
 0683 
 
 0708 
 
 0732 
 
 0757 
 
 0781 
 
 0806 
 
 0830 
 
 25-24 
 
 4.09 
 
 1.4 0854 
 
 0879 
 
 0903 
 
 0928 
 
 0952 
 
 0977 
 
 1001 
 
 1025 
 
 1050 
 
 1074 
 
 24 
 
 4.10 
 
 1.4 1099 
 
 1123 
 
 1147 
 
 1172 
 
 1196 
 
 1221 
 
 1245 
 
 1269 
 
 1294 
 
 1318 
 
 24 
 
 4.11 
 
 1.4 1342 
 
 1367 
 
 1391 
 
 1415 
 
 1440 
 
 1464 
 
 1488 
 
 1512 
 
 1537 
 
 1561 
 
 24 
 
 4.12 
 
 1.4 1585 
 
 1610 
 
 1634 
 
 1658 
 
 1682 
 
 1707 
 
 1731 
 
 1755 
 
 1779 
 
 1804 
 
 24 
 
 4.13 
 
 1.4 1828 
 
 1852 
 
 1876 
 
 1900 
 
 1925 
 
 1949 
 
 1973 
 
 1997 
 
 2021 
 
 2045 
 
 24 
 
 4.14 
 
 1.4 2070 
 
 2094 
 
 2118 
 
 2142 
 
 2166 
 
 2190 
 
 2214 
 
 2239 
 
 2263 
 
 2287 
 
 24 
 
 4.15 
 
 1.4 2311 
 
 2335 
 
 2359 
 
 2383 
 
 2407 
 
 2431 
 
 2455 
 
 2479 
 
 2503 
 
 2527 
 
 24 
 
 4.16 
 
 1.4 2552 
 
 2576 
 
 2600 
 
 2624 
 
 2648 
 
 2672 
 
 2696 
 
 2720 
 
 2744 
 
 2768 
 
 24 
 
 4.17 
 
 1.4 2792 
 
 2816 
 
 2840 
 
 2864 
 
 2887 
 
 2911 
 
 2935 
 
 2959 
 
 2983 
 
 3007 
 
 24 
 
 4.18 
 
 1.4 3031 
 
 3055 
 
 3079 
 
 3103 
 
 3127 
 
 3151 
 
 3175 
 
 3198 
 
 3222 
 
 3246 
 
 24 
 
 4.19 
 
 1.4 3270 
 
 3294 
 
 3318 
 
 3342 
 
 3365 
 
 3389 
 
 3413 
 
 3437 
 
 3461 
 
 3485 
 
 24 
 
 4.20 
 
 1.4 3508 
 
 3532 
 
 3556 
 
 3580 
 
 3604 
 
 3627 
 
 3651 
 
 3675 
 
 3699 
 
 3723 
 
 24 
 
 4.21 
 
 1.4 3746 
 
 3770 
 
 3794 
 
 3817 
 
 3841 
 
 3865 
 
 3889 
 
 3912 
 
 3936 
 
 3960 
 
 24 
 
 4.22 
 
 1.4 3984 
 
 4007 
 
 4031 
 
 4055 
 
 4078 
 
 4102 
 
 4126 
 
 4149 
 
 4173 
 
 4197 
 
 24 
 
 4.23 
 
 1.4 4220 
 
 4244 
 
 4267 
 
 4291 
 
 4315 
 
 4338 
 
 4362 
 
 4386 
 
 4409 
 
 4433 
 
 24 
 
 4.24 
 
 1.4 4456 
 
 4480 
 
 4503 
 
 4527 
 
 4551 
 
 4574 
 
 4598 
 
 4621 
 
 4645 
 
 4668 
 
 24 
 
 4.25 
 
 1.4 4692 
 
 4715 
 
 4739 
 
 4762 
 
 4786 
 
 4809 
 
 4833 
 
 4856 
 
 4880 
 
 4903 
 
 24-23 
 
 4.26 
 
 1.4 4927 
 
 4950 
 
 4974 
 
 4997 
 
 5021 
 
 5044 
 
 5068 
 
 5091 
 
 5115 
 
 5138 
 
 23 
 
 4.27 
 
 1.4 5161 
 
 5185 
 
 5208 
 
 5232 
 
 5255 
 
 5278 
 
 5302 
 
 5325 
 
 5349 
 
 5372 
 
 23 
 
 4.28 
 
 1.4 5395 
 
 5419 
 
 5442 
 
 5465 
 
 5489 
 
 5512 
 
 5535 
 
 5559 
 
 5582 
 
 5605 
 
 23 
 
 4.29 
 
 1.4 5629 
 
 5652 
 
 5675 
 
 5699 
 
 5722 
 
 5745 
 
 5768 
 
 5792 
 
 5815 
 
 5838 
 
 23 
 
 4.30 
 
 1.4 5862 
 
 5885 
 
 5908 
 
 5931 
 
 5954 
 
 5978 
 
 6001 
 
 6024 
 
 6047 
 
 6071 
 
 23 
 
 4.31 
 
 1.4 6094 
 
 6117 
 
 6140 
 
 6163 
 
 6187 
 
 6210 
 
 6233 
 
 6256 
 
 6279 
 
 6302 
 
 23 
 
 4.32 
 
 1.4 6326 
 
 6349 
 
 6372 
 
 6395 
 
 6418 
 
 6441 
 
 6464 
 
 6487 
 
 6511 
 
 6534 
 
 23 
 
 4.33 
 
 1.4 6557 
 
 6580 
 
 6603 
 
 6626 
 
 6649 
 
 6672 
 
 6695 
 
 6718 
 
 6741 
 
 6764 
 
 23 
 
 4.34 
 
 1.4 6787 
 
 6810 
 
 6834 
 
 6857 
 
 6880 
 
 6903 
 
 6926 
 
 6949 
 
 6972 
 
 6995 
 
 23 
 
 4.35 
 
 1.4 7018 
 
 7041 
 
 7064 
 
 7087 
 
 7109 
 
 7132 
 
 7155 
 
 7178 
 
 7201 
 
 7224 
 
 23 
 
 4.36 
 
 1.4 7247 
 
 7270 
 
 7293 
 
 7316 
 
 7339 
 
 7362 
 
 7385 
 
 7408 
 
 7431 
 
 7453 
 
 23 
 
 4.37 
 
 1.4 7476 
 
 7499 
 
 7522 
 
 7545 
 
 7568 
 
 7591 
 
 7614 
 
 7636 
 
 7659 
 
 7682 
 
 23 
 
 4.38 
 
 1.4 7705 
 
 7728 
 
 7751 
 
 7773 
 
 7796 
 
 7819 
 
 7842 
 
 7865 
 
 7887 
 
 7910 
 
 23 
 
 4.39 
 
 1.4 7933 
 
 7956 
 
 7978 
 
 8001 
 
 8024 
 
 8047 
 
 8070 
 
 8092 
 
 8115 
 
 8138 
 
 23 
 
 4.40 
 
 1.4 8160 
 
 8183 
 
 8206 
 
 8229 
 
 8251 
 
 8274 
 
 8297 
 
 8319 
 
 8342 
 
 8365 
 
 23 
 
 4.41 
 
 1.4 8387 
 
 8410 
 
 8433 
 
 8455 
 
 8478 
 
 8501 
 
 8523 
 
 8546 
 
 8569 
 
 8591 
 
 23 
 
 4.42 
 
 1.4 8614 
 
 8637 
 
 8659 
 
 8682 
 
 8704 
 
 8727 
 
 8750 
 
 8772 
 
 8795 
 
 8817 
 
 23 
 
 4.43 
 
 1.4 8840 
 
 8863 
 
 8885 
 
 8908 
 
 8930 
 
 8953 
 
 8975 
 
 8998 
 
 9020 
 
 9043 
 
 23 
 
 4.44 
 
 1.4 9065 
 
 9088 
 
 9110 
 
 9133 
 
 9155 
 
 9178 
 
 9200 
 
 9223 
 
 9245 
 
 9268 
 
 23 
 
 4.45 
 
 1.4 9290 
 
 9313 
 
 9335 
 
 9358 
 
 9380 
 
 9403 
 
 9425 
 
 9448 
 
 9470 
 
 9492 
 
 23-22 
 
 4.46 
 
 1.4 9515 
 
 9537 
 
 9560 
 
 9582 
 
 9605 
 
 9627 
 
 9649 
 
 9672 
 
 9694 
 
 9716 
 
 22 
 
 4.47 
 
 1.4 9739 
 
 9761 
 
 9784 
 
 9806 
 
 9828 
 
 9851 
 
 9873 
 
 9895 
 
 9918 
 
 9940 
 
 22 
 
 4.48 
 
 1.4 9962 
 
 9985 
 
 *0007 
 
 0029 
 
 0052 
 
 0074 
 
 0096 
 
 0118 
 
 0141 
 
 0163 
 
 22 
 
 4.49 
 
 1.5 0185 
 
 0208 
 
 0230 
 
 0252 
 
 0274 
 
 0297 
 
 0319 
 
 0341 
 
 0363 
 
 0386 
 
 22 
 
 4.50 
 
 1.5 0408 
 
 0430 
 
 0452 
 
 0474 
 
 0497 
 
 0519 
 
 0541 
 
 0563 
 
 0585 
 
 0608 
 
 22 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
TABLES. 
 Five-Place Natural Logarithms. 
 
 137 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 4.50 
 
 1.5 0408 
 
 0430 
 
 0452 
 
 0474 
 
 0497 
 
 0519 
 
 0541 
 
 0563 
 
 0585 
 
 0608 
 
 22 
 
 4.51 
 
 1.5 0630 
 
 0652 
 
 0674 
 
 0696 
 
 0718 
 
 0741 
 
 0763 
 
 0785 
 
 0807 
 
 0829 
 
 22 
 
 4.52 
 
 1.5 0851 
 
 0873 
 
 0895 
 
 0918 
 
 0940 
 
 0962 
 
 0984 
 
 1006 
 
 1028 
 
 1050 
 
 22 
 
 4.53 
 
 1.5 1072 
 
 1094 
 
 1116 
 
 1138 
 
 1160 
 
 1183 
 
 1205 
 
 1227 
 
 1249 
 
 1271 
 
 22 
 
 4.54 
 
 1.5 1293 
 
 1315 
 
 1337 
 
 1359 
 
 1381 
 
 1403 
 
 1425 
 
 1447 
 
 1469 
 
 1491 
 
 22 
 
 4.55 
 
 1.5 1513 
 
 1535 
 
 1557 
 
 1579 
 
 1601 
 
 1623 
 
 1645 
 
 1666 
 
 1688 
 
 1710 
 
 22 
 
 4.56 
 
 1.5 1732 
 
 1754 
 
 1776 
 
 1798 
 
 1820 
 
 1842 
 
 1864 
 
 1886 
 
 1908 
 
 1929 
 
 22 
 
 4.57 
 
 1.5 1951 
 
 1973 
 
 1995 
 
 2017 
 
 2039 
 
 2061 
 
 2083 
 
 2104 
 
 2126 
 
 2148 
 
 22 
 
 4.58 
 
 1.5 2170 
 
 2192 
 
 2214 
 
 2235 
 
 2257 
 
 2279 
 
 2301 
 
 2323 
 
 2344 
 
 2366 
 
 22 
 
 4.59 
 
 1.5 2388 
 
 2410 
 
 2432 
 
 2453 
 
 2475 
 
 2497 
 
 2519 
 
 2540 
 
 2562 
 
 2584 
 
 22 
 
 4.60 
 
 1.5 2606 
 
 2627 
 
 2649 
 
 2671 
 
 2693 
 
 2714 
 
 2736 
 
 2758 
 
 2779 
 
 2801 
 
 22 
 
 4.61 
 
 1.5 2823 
 
 2844 
 
 2866 
 
 2888 
 
 2910 
 
 2931 
 
 2953 
 
 2975 
 
 2996 
 
 3018 
 
 22 
 
 4.62 
 
 1.5 3039 
 
 3061 
 
 3083 
 
 3104 
 
 3126 
 
 3148 
 
 3169 
 
 3191 
 
 3212 
 
 3234 
 
 22 
 
 4.63 
 
 1.5 3256 
 
 3277 
 
 3299 
 
 3320 
 
 3342 
 
 3364 
 
 3385 
 
 3407 
 
 3428 
 
 3450 
 
 22 
 
 4.64 
 
 1.5 3471 
 
 3493 
 
 3515 
 
 3536 
 
 3558 
 
 3579 
 
 3601 
 
 3622 
 
 3644 
 
 3665 
 
 22 
 
 4.65 
 
 1.5 3687 
 
 3708 
 
 3730 
 
 3751 
 
 3773 
 
 3794 
 
 3816 
 
 3837 
 
 3859 
 
 3880 
 
 22-21 
 
 4.66 
 
 1.5 3902 
 
 3923 
 
 3944 
 
 3966 
 
 3987 
 
 4009 
 
 4030 
 
 4052 
 
 4073 
 
 4094 
 
 21 
 
 4.67 
 
 1.5 4116 
 
 4137 
 
 4159 
 
 4180 
 
 4202 
 
 4223 
 
 4244 
 
 4266 
 
 4287 
 
 4308 
 
 21 
 
 4.68 
 
 1.5 4330 
 
 4351 
 
 4373 
 
 4394 
 
 4415 
 
 4437 
 
 4458 
 
 4479 
 
 4501 
 
 4522 
 
 21 
 
 4.69 
 
 1.5 4543 
 
 4565 
 
 4586 
 
 4607 
 
 4629 
 
 4650 
 
 4671 
 
 4692 
 
 4714 
 
 4735 
 
 21 
 
 4.70 
 
 1.5 4756 
 
 4778 
 
 4799 
 
 4820 
 
 4841 
 
 4863 
 
 4884 
 
 4905 
 
 4926 
 
 4948 
 
 21 
 
 4.71 
 
 1.5 4969 
 
 4990 
 
 5011 
 
 5032 
 
 5054 
 
 5075 
 
 5096 
 
 5117 
 
 5138 
 
 5160 
 
 21 
 
 4.72 
 
 1.5 5181 
 
 5202 
 
 5223 
 
 5244 
 
 5266 
 
 5287 
 
 5308 
 
 5329 
 
 5350 
 
 5371 
 
 21 
 
 4.73 
 
 1.5 5393 
 
 5414 
 
 5435 
 
 5456 
 
 5477 
 
 5498 
 
 5519 
 
 5540 
 
 5562 
 
 5583 
 
 21 
 
 4.74 
 
 1.5 5604 
 
 5625 
 
 5646 
 
 5667 
 
 5688 
 
 5709 
 
 5730 
 
 5751 
 
 5772 
 
 5793 
 
 21 
 
 4.75 
 
 1.5 5814 
 
 5836 
 
 5857 
 
 5878 
 
 5899 
 
 5920 
 
 5941 
 
 5962 
 
 5983 
 
 6004 
 
 21 
 
 4.76 
 
 1.5 6025 
 
 6046 
 
 6067 
 
 6088 
 
 6109 
 
 6130 
 
 6151 
 
 6172 
 
 6193 
 
 6214 
 
 21 
 
 4.77 
 
 1.5 6235 
 
 6256 
 
 6277 
 
 6298 
 
 6318 
 
 6339 
 
 6360 
 
 6381 
 
 6402 
 
 6423 
 
 21 
 
 4.78 
 
 1.5 6444 
 
 6465 
 
 6486 
 
 6507 
 
 6528 
 
 6549 
 
 6569 
 
 6590 
 
 6611 
 
 6632 
 
 21 
 
 4.79 
 
 1.5 6653 
 
 6674 
 
 6695 
 
 6716 
 
 6737 
 
 6757 
 
 6778 
 
 6799 
 
 6820 
 
 6841 
 
 21 
 
 4.80 
 
 1.5 6862 
 
 6882 
 
 6903 
 
 6924 
 
 6945 
 
 6966 
 
 6987 
 
 7007 
 
 7028 
 
 7049 
 
 21 
 
 4.81 
 
 1.5 7070 
 
 7090 
 
 7111 
 
 7132 
 
 7153 
 
 7174 
 
 7194 
 
 7215 
 
 7236 
 
 7257 
 
 21 
 
 4.82 
 
 1.5 7277 
 
 7298 
 
 7319 
 
 7340 
 
 7360 
 
 7381 
 
 7402 
 
 7423 
 
 7443 
 
 7464 
 
 21 
 
 4.83 
 
 1.5 7485 
 
 7505 
 
 7526 
 
 7547 
 
 7567 
 
 7588 
 
 7609 
 
 7629 
 
 7650 
 
 7671 
 
 21 
 
 4.84 
 
 1.5 7691 
 
 7712 
 
 7733 
 
 7753 
 
 7774 
 
 7795 
 
 7815 
 
 7836 
 
 7857 
 
 7877 
 
 21 
 
 4.85 
 
 1.5 7898 
 
 7918 
 
 7939 
 
 7960 
 
 7980 
 
 8001 
 
 8022 
 
 8042 
 
 8063 
 
 8083 
 
 21 
 
 4.86 
 
 1.5 8104 
 
 8124 
 
 8145 
 
 8166 
 
 8186 
 
 8207 
 
 8227 
 
 8248 
 
 8268 
 
 8289 
 
 21 
 
 4.87 
 
 1.5 8309 
 
 8330 
 
 8350 
 
 8371 
 
 8391 
 
 8412 
 
 8433 
 
 8453 
 
 8474 
 
 8494 
 
 21-20 
 
 4.88 
 
 1.5 8515 
 
 8535 
 
 8555 
 
 8576 
 
 8596 
 
 8617 
 
 8637 
 
 8658 
 
 8678 
 
 8699 
 
 20 
 
 4.89 
 
 1.5 8719 
 
 8740 
 
 8760 
 
 8781 
 
 8S01 
 
 8821 
 
 8842 
 
 8862 
 
 8883 
 
 8903 
 
 20 
 
 4.90 
 
 1.5 8924 
 
 8944 
 
 8964 
 
 8985 
 
 9005 
 
 9026 
 
 9046 
 
 9066 
 
 9087 
 
 9107 
 
 20 
 
 4.91 
 
 1.5 9127 
 
 9148 
 
 9168 
 
 9188 
 
 9209 
 
 9229 
 
 9250 
 
 9270 
 
 9290 
 
 9311 
 
 20 
 
 4.92 
 
 1.5 9331 
 
 9351 
 
 9371 
 
 9392 
 
 9412 
 
 9432 
 
 9453 
 
 9473 
 
 9493 
 
 9514 
 
 20 
 
 4.93 
 
 1.5 9534 
 
 955' 
 
 9574 
 
 9595 
 
 9615 
 
 9635 
 
 9656 
 
 9676 
 
 9696 
 
 9716 
 
 20 
 
 4.94 
 
 1.5 9737 
 
 9757 
 
 9777 
 
 9797 
 
 9817 
 
 9838 
 
 9858 
 
 9878 
 
 9898 
 
 9919 
 
 20 
 
 4.95 
 
 1.5 9939 
 
 9959 
 
 9979 
 
 9999 *0020 
 
 0040 
 
 0060 
 
 OOSO 
 
 0100 
 
 0120 
 
 20 
 
 4.96 
 
 1.6 0141 
 
 0161 
 
 0181 
 
 0201 
 
 0221 
 
 0241 
 
 0261 
 
 028? 
 
 0302 
 
 0322 
 
 20 
 
 4.97 
 
 1.6 0342 
 
 0362 
 
 0382 
 
 0402 
 
 0422 
 
 0443 
 
 0463 
 
 0483 
 
 0503 
 
 0523 
 
 20 
 
 4.98 
 
 1.6 0543 
 
 0563 
 
 0583 
 
 0603 
 
 0623 
 
 0643 
 
 0663 
 
 0683 
 
 0704 
 
 0724 
 
 20 
 
 4.99 
 
 1.6 0744 
 
 0764 
 
 0784 
 
 0804 
 
 0824 
 
 0844 
 
 0864 
 
 0884 
 
 0904 
 
 0924 
 
 20 
 
 5>00 
 
 1.6 0944 
 
 0964 
 
 0984 
 
 1004 
 
 1024 
 
 1044 
 
 1064 
 
 1084 
 
 1104 
 
 1124 
 
 20 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
138 
 
 TABLES. 
 
 Five-Place Natural Logarithms. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 5.0 
 
 1.6 0944 
 
 1144 
 
 1343 
 
 1542 
 
 1741 
 
 1939 
 
 2137 
 
 2334 
 
 2531 
 
 2728 
 
 200-196 
 
 5.1 
 
 1.6 2924 
 
 3120 
 
 3315 
 
 3511 
 
 3705 
 
 3900 
 
 4094 
 
 4287 
 
 44S1 
 
 4673 
 
 196-192 
 
 5.2 
 
 1.6 4866 
 
 5058 
 
 5250 
 
 5441 
 
 5632 
 
 5823 
 
 6013 
 
 6203 
 
 6393 
 
 6582 
 
 192-189 
 
 5.3 
 
 1.6 6771 
 
 6959 
 
 7147 
 
 7335 
 
 7523 
 
 7710 
 
 7896 
 
 8083 
 
 8269 
 
 8455 
 
 189-185 
 
 5.4 
 
 1.6 8640 
 
 8825 
 
 9010 
 
 9194 
 
 9378 
 
 9562 
 
 9745 
 
 9928 *0111 
 
 0293 
 
 185-182 
 
 5.5 
 
 1.7 0475 
 
 0656 
 
 0838 
 
 1019 
 
 1199 
 
 1380 
 
 1560 
 
 1740 
 
 1919 
 
 2098 
 
 182-179 
 
 5.6 
 
 1.7 2277 
 
 2455 
 
 2633 
 
 2811 
 
 2988 
 
 3166 
 
 3342 
 
 3519 
 
 3695 
 
 3871 
 
 178-176 
 
 5.7 
 
 1.7 4047 
 
 4222 
 
 4397 
 
 4572 
 
 4746 
 
 4920 
 
 5094 
 
 5267 
 
 5440 
 
 5613 
 
 175-173 
 
 5.8 
 
 1.7 5786 
 
 5958 
 
 6130 
 
 6302 
 
 6473 
 
 6644 
 
 6815 
 
 6985 
 
 7156 
 
 7326 
 
 172-170 
 
 5.9 
 
 1.7 7495 
 
 7665 
 
 7834 
 
 8002 
 
 8171 
 
 8339 
 
 8507 
 
 8675 
 
 8842 
 
 9009 
 
 169-167 
 
 6.0 
 
 1.7 9176 
 
 9342 
 
 9509 
 
 9675 
 
 9840 
 
 *0006 
 
 0171 
 
 0336 
 
 0500 
 
 0665 
 
 167-164 
 
 6.1 
 
 1.8 0829 
 
 0993 
 
 1156 
 
 1319 
 
 1482 
 
 1645 
 
 1808 
 
 1970 
 
 2132 
 
 2294 
 
 164-161 
 
 6.2 
 
 1.8 2455 
 
 2616 
 
 2777 
 
 2938 
 
 3098 
 
 3258 
 
 3418 
 
 3578 
 
 3737 
 
 3896 
 
 161-159 
 
 6.3 
 
 1.8 4055 
 
 4214 
 
 4372 
 
 4530 
 
 4688 
 
 4845 
 
 5003 
 
 5160 
 
 5317 
 
 5473 
 
 159-156 
 
 6.4 
 
 1.8 5630 
 
 5786 
 
 5942 
 
 6097 
 
 6253 
 
 6408 
 
 6563 
 
 6718 
 
 6872 
 
 7026 
 
 156-154 
 
 6.5 
 
 1.8 7180 
 
 7334 
 
 7487 
 
 7641 
 
 7794 
 
 7947 
 
 8099 
 
 8251 
 
 8403 
 
 8555 
 
 154-152 
 
 6.6 
 
 1.8 8707 
 
 8858 
 
 9010 
 
 9160 
 
 931] 
 
 9462 
 
 9612 
 
 9762 
 
 9912 *0061 
 
 151-149 
 
 6.7 
 
 1.9 0211 
 
 0360 
 
 0509 
 
 0658 
 
 0806 
 
 0954 
 
 1102 
 
 1250 
 
 1398 
 
 1545 
 
 149-147 
 
 6.8 
 
 1.9 1692 
 
 1839 
 
 1986 
 
 2132 
 
 2279 
 
 2425 
 
 2571 
 
 2716 
 
 2862 
 
 3007 
 
 147-145 
 
 6.9 
 
 1.9 3152 
 
 3297 
 
 3442 
 
 3586 
 
 3730 
 
 3874 
 
 4018 
 
 4162 
 
 4305 
 
 4448 
 
 145-143 
 
 7.0 
 
 1.9 4591 
 
 4734 
 
 4876 
 
 5019 
 
 5161 
 
 5303 
 
 5445 
 
 5586 
 
 5727 
 
 5869 
 
 143-141 
 
 7.1 
 
 1.9 6009 
 
 6150 
 
 6291 
 
 6431 
 
 6571 
 
 6711 
 
 6851 
 
 6991 
 
 7130 
 
 7269 
 
 141-139 
 
 7.2 
 
 1.9 7408 
 
 7547 
 
 7685 
 
 7824 
 
 7962 
 
 8100 
 
 8238 
 
 8376 
 
 8513 
 
 8650 
 
 139-137 
 
 7.3 
 
 1.9 8787 
 
 8924 
 
 9061 
 
 9198 
 
 9334 
 
 9470 
 
 9606 
 
 9742 
 
 9877 *0013 
 
 137-135 
 
 7.4 
 
 2.0 0148 
 
 0283 
 
 0418 
 
 0553 
 
 0687 
 
 0821 
 
 0956 
 
 1089 
 
 1223 
 
 1357 
 
 135-133 
 
 7.5 
 
 2.0 1490 
 
 1624 
 
 1757 
 
 1890 
 
 2022 
 
 2155 
 
 2287 
 
 2419 
 
 2551 
 
 2683 
 
 133-132 
 
 7.6 
 
 2.0 2815 
 
 2946 
 
 3078 
 
 3209 
 
 3340 
 
 3471 
 
 3601 
 
 3732 
 
 3862 
 
 3992 
 
 131-130 
 
 7.7 
 
 2.0 4122 
 
 4252 
 
 4381 
 
 4511 
 
 4640 
 
 4769 
 
 4898 
 
 5027 
 
 5156 
 
 5284 
 
 130-128 
 
 7.8 
 
 2.0 5412 
 
 5540 
 
 5668 
 
 5796 
 
 5924 
 
 6051 
 
 6179 
 
 6306 
 
 6433 
 
 6560 
 
 128-127 
 
 7.9 
 
 2.0 6686 
 
 6813 
 
 6939 
 
 7065 
 
 7191 
 
 7317 
 
 7443 
 
 7568 
 
 7694 
 
 7819 
 
 127-125 
 
 8.0 
 
 2.0 7944 
 
 8069 
 
 8194 
 
 8318 
 
 8443 
 
 8567 
 
 8691 
 
 8815 
 
 8939 
 
 9063 
 
 125-124 
 
 8.1 
 
 2.0 9186 
 
 9310 
 
 9433 
 
 9556 
 
 9679 
 
 9802 
 
 9924 *0047 
 
 0169 
 
 0291 
 
 123-122 
 
 8.2 
 
 2.1 0413 
 
 0535 
 
 0657 
 
 0779 
 
 0900 
 
 1021 
 
 1142 
 
 1263 
 
 1384 
 
 1505 
 
 122-121 
 
 8.3 
 
 2.1 1626 
 
 1746 
 
 1866 
 
 1986 
 
 2106 
 
 2226 
 
 2346 
 
 2465 
 
 2585 
 
 2704 
 
 120-119 
 
 8.4 
 
 2.1 2823 
 
 2942 
 
 3061 
 
 3180 
 
 3298 
 
 3417 
 
 3535 
 
 3653 
 
 3771 
 
 3889 
 
 119-118 
 
 8.5 
 
 2.1 4007 
 
 4124 
 
 4242 
 
 4359 
 
 4476 
 
 4593 
 
 4710 
 
 4827 
 
 4943 
 
 5060 
 
 118-116 
 
 8.6 
 
 2.1 5176 
 
 5292 
 
 5409 
 
 5524 
 
 5640 
 
 5756 
 
 5871 
 
 5987 
 
 6102 
 
 6217 
 
 116-115 
 
 8.7 
 
 2.1 6332 
 
 6447 
 
 6562 
 
 6677 
 
 6791 
 
 6905 
 
 7020 
 
 7134 
 
 7248 
 
 7361 
 
 115-114 
 
 8.8 
 
 2.1 7475 
 
 7589 
 
 7702 
 
 7816 
 
 7929 
 
 8042 
 
 8155 
 
 8267 
 
 8380 
 
 8493 
 
 114-112 
 
 8.9 
 
 2.1 8605 
 2.1 9722 
 
 8717 
 9834 
 
 8830 8942 
 9944 *0055 
 
 9054 
 0166 
 
 9165 
 
 9277 
 
 9389 
 
 9500 
 
 9611 
 
 112-111 
 
 9.0 
 
 0276 
 
 0387 
 
 0497 
 
 0607 
 
 0717 
 
 111-110 
 
 9.1 
 
 2.2 0827 
 
 0937 
 
 1047 
 
 1157 
 
 1266 
 
 1375 
 
 1485 
 
 1594 
 
 1703 
 
 1812 
 
 110-109 
 
 9.2 
 
 2.2 1920 
 
 2029 
 
 2138 
 
 2246 
 
 2354 
 
 2462 
 
 2570 
 
 2678 
 
 2786 
 
 2894 
 
 109-108 
 
 9.3 
 
 2.2 3001 
 
 3109 
 
 3216 
 
 3324 
 
 3431 
 
 3538 
 
 3645 
 
 3751 
 
 3858 
 
 3965 
 
 107-106 
 
 9.4 
 
 2.2 4071 
 
 4177 
 
 4284 
 
 4390 
 
 4496 
 
 4601 
 
 4707 
 
 4813 
 
 4918 
 
 5024 
 
 106-105 
 
 9.5 
 
 2.2 5129 
 
 5234 
 
 5339 
 
 5444 
 
 5549 
 
 5654 
 
 5759 
 
 5863 
 
 5968 
 
 6072 
 
 105-104 
 
 9.6 
 
 2.2 6176 
 
 6280 
 
 6384 
 
 6488 
 
 6592 
 
 6696 
 
 6799 
 
 6903 
 
 7006 
 
 7109 
 
 104-103 
 
 9.7 
 
 2.2 7213 
 
 7316 
 
 7419 
 
 7521 
 
 7624 
 
 7727 
 
 7829 
 
 7932 
 
 8034 
 
 8136 
 
 103-102 
 
 9.8 
 
 2.2 8238 
 
 8340 
 
 8442 
 
 8544 
 
 8646 
 
 8747 
 
 8849 
 
 8950 
 
 9051 
 
 9152 
 
 102-101 
 
 9.9 
 
 2.2 9253 
 
 9354 
 
 9455 
 
 9556 
 
 9657 
 
 9757 
 
 9858 
 
 9958 *0058 
 
 0158 
 
 101-100 
 
 10.0 
 
 2.3 0259 
 
 0358 
 
 0458 
 
 0558 
 
 0658 
 
 0757 
 
 0857 
 
 0956 
 
 1055 
 
 1154 
 
 100-99 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
TABLES. 139 
 
 The Natural Logarithms (each increased by 10.) of Numbers between 0.00 and 0.99. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 0.0 
 
 
 5.395 
 
 6.088 
 
 6.493 
 
 6.781 
 
 7.004 
 
 7.187 
 
 7.341 
 
 7.474 
 
 7.592 
 
 0.1 
 
 7.697 
 
 7.793 
 
 1880 
 
 7.960 
 
 8.034 
 
 8.103 
 
 8.167 
 
 8.228 
 
 8.285 
 
 8.339 
 
 0.2 
 
 8.391 
 
 8.439 
 
 8.486 
 
 8.530 
 
 8.573 
 
 8.614 
 
 8.653 
 
 8.691 
 
 8.727 
 
 8.762 
 
 0.3 
 
 8.796 
 
 8.829 
 
 8.861 
 
 8.891 
 
 8.921 
 
 8.950 
 
 8.978 
 
 9.006 
 
 9.032 
 
 9.058 
 
 0.4 
 
 9.084 
 
 9.10S 
 
 9.132 
 
 9.156 
 
 9.179 
 
 9.201 
 
 9.223 
 
 9.245 
 
 9.266 
 
 9.287 
 
 0.5 
 
 9.307 
 
 9.327 
 
 9.346 
 
 9.365 
 
 9.384 
 
 9.402 
 
 9.420 
 
 9.438 
 
 9.455 
 
 9.472 
 
 0.6 
 
 9.489 
 
 9.506 
 
 9.522 
 
 9.538 
 
 9.554 
 
 9.569 
 
 9.584 
 
 9.600 
 
 9.614 
 
 9.629 
 
 0.7 
 
 9.643 
 
 9.658 
 
 9.671 
 
 9.685 
 
 9.699 
 
 9.712 
 
 9.726 
 
 9.739 
 
 9.752 
 
 9.764 
 
 0.8 
 
 9.777 
 
 9.789 
 
 9.802 
 
 9.814 
 
 9.826 
 
 9.837 
 
 9.849 
 
 9.861 
 
 9.872 
 
 9.883 
 
 0.9 
 
 9.895 
 
 9.906 
 
 9.917 
 
 9.927 
 
 9.938 
 
 9.949 
 
 9.959 
 
 9.970 
 
 9.980 
 
 9.990 
 
 Note : loggX = logioX • lege 10 =: (2.30259) logio x. 
 
 The Natural Logarithms of Whole Numbers from 10 to 209. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
 2.3026 
 
 3979 
 
 4849 
 
 5649 
 
 6391 
 
 7080 
 
 7726 
 
 8332 
 
 8904 
 
 9444 
 
 2 
 
 2.9957 
 
 *0445 
 
 0910 
 
 1355 
 
 1781 
 
 2189 
 
 2581 
 
 2958 
 
 3322 
 
 3673 
 
 3 
 
 3.4012 
 
 4340 
 
 4657 
 
 4965 
 
 5264 
 
 5553 
 
 5835 
 
 6109 
 
 6376 
 
 6636 
 
 4 
 
 3.6889 
 
 7136 
 
 7377 
 
 7612 
 
 7842 
 
 8067 
 
 8286 
 
 8501 
 
 8712 
 
 8918 
 
 5 
 
 3.9120 
 
 9318 
 
 9512 
 
 9703 
 
 9890 
 
 *0073 
 
 0254 
 
 0431 
 
 0604 
 
 0775 
 
 6 
 
 4.0943 
 
 1109 
 
 1271 
 
 1431 
 
 1589 
 
 1744 
 
 1897 
 
 2047 
 
 2195 
 
 2341 
 
 7 
 
 4.2485 
 
 2627 
 
 2767 
 
 2905 
 
 3041 
 
 3175 
 
 3307 
 
 3438 
 
 3567 
 
 3694 
 
 8 
 
 4.3820 
 
 3944 
 
 4067 
 
 4188 
 
 4308 
 
 4427 
 
 4543 
 
 4659 
 
 4773 
 
 4886 
 
 9 
 
 4.4998 
 
 5109 
 
 5218 
 
 5326 
 
 5433 
 
 5539 
 
 5643 
 
 5747 
 
 5850 
 
 5951 
 
 10 
 
 4.6052 
 
 6151 
 
 6250 
 
 6347 
 
 6444 
 
 6540 
 
 6634 
 
 6728 
 
 6821 
 
 6913 
 
 11 
 
 4.7005 
 
 7095 
 
 7185 
 
 7274 
 
 7362 
 
 7449 
 
 7536 
 
 7622 
 
 7707 
 
 7791 
 
 12 
 
 4.7875 
 
 7958 
 
 8040 
 
 8122 
 
 8203 
 
 8283 
 
 8363 
 
 8442 
 
 8520 
 
 8598 
 
 13 
 
 4.8675 
 
 8752 
 
 8828 
 
 8903 
 
 8978 
 
 9053 
 
 9127 
 
 9200 
 
 9273 
 
 9345 
 
 14 
 
 4.9416 
 
 94SS 
 
 9558 
 
 9628 
 
 9698 
 
 9767 
 
 9836 
 
 9904 
 
 9972 
 
 *0039 
 
 15 
 
 5.0106 
 
 0173 
 
 0239 
 
 0304 
 
 0370 
 
 0434 
 
 0499 
 
 0562 
 
 0626 
 
 0689 
 
 16 
 
 5.0752 
 
 0814 
 
 0876 
 
 0938 
 
 0999 
 
 1059 
 
 1120 
 
 1180 
 
 1240 
 
 1299 
 
 17 
 
 5.1358 
 
 P17 
 
 1475 
 
 1533 
 
 1591 
 
 1648 
 
 1705 
 
 1762 
 
 1818 
 
 1874 
 
 18 
 
 5.1930 
 
 1985 
 
 2040 
 
 2095 
 
 2149 
 
 2204 
 
 2257 
 
 2311 
 
 2364 
 
 2417 
 
 19 
 
 5.2470 
 
 2523 
 
 2575 
 
 2627 
 
 2679 
 
 2730 
 
 2781 
 
 2832 
 
 2883 
 
 2933 
 
 20 
 
 5.2983 
 
 3033 
 
 3083 
 
 3132 
 
 3181 
 
 3230 
 
 3279 
 
 3327 
 
 3375 
 
 3423 
 
 Note : loge 10 = 2,30258509. 
 
 lege 100 = 4.60517019. 
 
140 
 
 TABLES. 
 
 The Common Logarithms of r (n) for Values of n between 1 and 2. 
 r(n)= j x"-i-e-^dx= j log- dx. 
 
 n 
 
 9^ 
 
 o 
 
 n 
 
 o 
 
 ho 
 
 O 
 
 r— » 
 
 n 
 
 3^ 
 
 o 
 
 n 
 
 3; 
 
 2 
 
 0^ 
 
 n 
 
 3; 
 
 bO 
 
 1.01 
 
 1.9975 
 
 1.21 
 
 T.9617 
 
 1.41 
 
 1.9478 
 
 1.61 
 
 1.9517 
 
 1.81 
 
 1.9704 
 
 1.02 
 
 1.9951 
 
 1.22 
 
 1.9605 
 
 1.42 
 
 1.9476 
 
 1.62 
 
 1.9523 
 
 1.82 
 
 1.9717 
 
 1.03 
 
 1.9928 
 
 1.23 
 
 1.9594 
 
 1.43 
 
 1.9475 
 
 1.63 
 
 1.9529 
 
 1.83 
 
 1.9730 
 
 1.04 
 
 1.9905 
 
 1.24 
 
 1.9583 
 
 1.44 
 
 1.9473 
 
 1.64 
 
 1.9536 
 
 1.84 
 
 1.9743 
 
 1.05 
 
 1.9883 
 
 1.25 
 
 1.9573 
 
 1.45 
 
 1.9473 
 
 1.65 
 
 1.9543 
 
 1.85 
 
 1.9757 
 
 1.06 
 
 T.9862 
 
 1.26 
 
 1.9564 
 
 1.46 
 
 1.9472 
 
 1.66 
 
 1.9550 
 
 1.86 
 
 1.9771 
 
 1.07 
 
 T.9841 
 
 1.27 
 
 T.9554 
 
 1.47 
 
 1.9473 
 
 1.67 
 
 1.9558 
 
 1.87 
 
 1.9786 
 
 l.OS 
 
 1.9821 
 
 1.28 
 
 1.9546 
 
 1.48 
 
 1.9473 
 
 1.68 
 
 T.9566 
 
 1.88 
 
 1.9800 
 
 1.09 
 
 1.9802 
 
 1.29 
 
 1.9538 
 
 1.49 
 
 1.9474 
 
 1.69 
 
 1.9575 
 
 1.89 
 
 1.9815 
 
 1.10 
 
 1.9783 
 
 1.30 
 
 1.9530 
 
 1.50 
 
 1.9475 
 
 1.70 
 
 T.9584 
 
 1.90 
 
 1.9831 
 
 1.11 
 
 1.9765 
 
 1.31 
 
 T.9523 
 
 1.51 
 
 1.9477 
 
 1.71 
 
 1.9593 
 
 1.91 
 
 T.9846 
 
 1.12 
 
 1.9748 
 
 1.32 
 
 1.9516 
 
 1.52 
 
 1.9479 
 
 1.72 
 
 1.9603 
 
 1.92 
 
 1.9862 
 
 1.13 
 
 1.9731 
 
 1.33 
 
 1.9510 
 
 1.53 
 
 1.9482 
 
 1.73 
 
 1.9613 
 
 1.93 
 
 1.9878 
 
 1.14 
 
 1.9715 
 
 1.34 
 
 T.9505 
 
 1.54 
 
 1.9485 
 
 1.74 
 
 T.9623 
 
 1.94 
 
 1.9895 
 
 1.15 
 
 1.9699 
 
 1.35 
 
 1.9500 
 
 1.55 
 
 1.9488 
 
 1.75 
 
 1.9633 
 
 1.95 
 
 1.9912 
 
 1.16 
 
 1.9684 
 
 1.36 
 
 1.9495 
 
 1.56 
 
 1.9492 
 
 1.76 
 
 1.9644 
 
 1.96 
 
 T.9929 
 
 1.17 
 
 1.9669 
 
 1.37 
 
 1.9491 
 
 1.57 
 
 1.9496 
 
 1.77 
 
 T.9656 
 
 1.97, 
 
 1.9946 
 
 1.18 
 
 1.9655 
 
 1.38 
 
 1.9487 
 
 1.5S 
 
 1.9501 
 
 1.78 
 
 T.9667 
 
 1.98 
 
 1.9964 
 
 1.19 
 
 1.9642 
 
 1.39 
 
 1.9483 
 
 1.59 
 
 1.9506 
 
 1.79 
 
 1.9679 
 
 1.99 
 
 1.9982 
 
 1.20 
 
 1.9629 
 
 1.40 
 
 1.9481 
 
 1.60 
 
 1.9511 
 
 1.80 
 
 1.9691 
 
 200 
 
 0.0000 
 
 r(2 + i) = z-r(2), z>i. 
 
TABLES. 
 
 141 
 
 NATURAL TRIGONOMETRIC FUNCTIONS. 
 
 Angle. 
 
 Sin. 
 
 Csc. 
 
 Tan. 
 
 Ctn, 
 
 Sec. 
 
 Cos. 
 
 
 0° 
 
 0.000 
 
 00 
 
 0.000 
 
 00 
 
 1.000 
 
 1.000 
 
 90° 
 
 1 
 
 0.017 
 
 57.30 
 
 0.017 
 
 57.29 
 
 1.000 
 
 1.000 
 
 89 
 
 2 
 
 0.035 
 
 28.65 
 
 0.035 
 
 28.64 
 
 1.001 
 
 0.999 
 
 88 
 
 3 
 
 0.052 
 
 19.11 
 
 0.052 
 
 19.08 
 
 1.001 
 
 0.999 
 
 87 
 
 4 
 
 0.070 
 
 14.34 
 
 0.070 
 
 14.30 
 
 1.002 
 
 0.998 
 
 86 
 
 5° 
 
 0.0S7 
 
 11.47 
 
 0.0S7 
 
 11.43 
 
 1.004 
 
 0.996 
 
 85° 
 
 6 
 
 0.105 
 
 9.567 
 
 0.105 
 
 9.514 
 
 1.006 
 
 0.995 
 
 84 
 
 7 
 
 0.122 
 
 8.206 
 
 0.123 
 
 8.144 
 
 1.008 
 
 0.993 
 
 83 
 
 8 
 
 0.139 
 
 7.185 
 
 0.141 
 
 7.115 
 
 1.010 
 
 0.990 
 
 82 
 
 9 
 
 0.156 
 
 6.392 
 
 0.158 
 
 6.314 
 
 1.012 
 
 0.988 
 
 81 
 
 10° 
 
 0.174 
 
 5.759 
 
 0.176 
 
 5.671 
 
 1.015 
 
 0.985 
 
 80° 
 
 11 
 
 0.191 
 
 5.241 
 
 0.194 
 
 5.145 
 
 1.019 
 
 0.982 
 
 79 
 
 12 
 
 0.208 
 
 4.810 
 
 0.213 
 
 4.705 
 
 1.022 
 
 0.978 
 
 78 
 
 13 
 
 0.225 
 
 4.445 
 
 0.231 
 
 4.331 
 
 1.026 
 
 0.974 
 
 77 
 
 14 
 
 0.242 
 
 4.134 
 
 0.249 
 
 4.011 
 
 1.031 
 
 0.970 
 
 76 
 
 15° 
 
 0.259 
 
 3.864 
 
 0.268 
 
 3.732 
 
 1.035 
 
 0.966 
 
 75° 
 
 16 
 
 0.276 
 
 3.628 
 
 0.287 
 
 3.487 
 
 1.040 
 
 0.961 
 
 74 
 
 17 
 
 0.292 
 
 3.420 
 
 0.306 
 
 3.271 
 
 1.046 
 
 0.956 
 
 73 
 
 18 
 
 0.309 
 
 3.236 
 
 0.325 
 
 3.078 
 
 1.051 
 
 0.951 
 
 72 
 
 19 
 
 0.326 
 
 3.072 
 
 0.344 
 
 2.904 
 
 1.058 
 
 0.946 
 
 71 
 
 20° 
 
 0.342 
 
 2.924 
 
 0.364 
 
 2.747 
 
 1.064 
 
 0.940 
 
 70° 
 
 21 
 
 0.358 
 
 2.790 
 
 0.384 
 
 2.605 
 
 1.071 
 
 0.934 
 
 69 
 
 22 
 
 0.375 
 
 2.669 
 
 0.404 
 
 2.475 
 
 1.079 
 
 0.927 
 
 68 
 
 23 
 
 0.391 
 
 2.559 
 
 0.424 
 
 2.356 
 
 1.086 
 
 0.921 
 
 67 
 
 24 
 
 0.407 
 
 2.459 
 
 0.445 
 
 2.246 
 
 1.095 
 
 0.914 
 
 66 
 
 25^ 
 
 0.423 
 
 2.366 
 
 0.466 
 
 2.145 
 
 1.103 
 
 0.906 
 
 65° 
 
 26 
 
 0.438 
 
 2.281 
 
 0.488 
 
 2.050 
 
 1.113 
 
 0.899 
 
 64 
 
 27 
 
 0.454 
 
 2.203 
 
 0.5 JO 
 
 1.963 
 
 1.122 
 
 0.891 
 
 63 
 
 28 
 
 0.469 
 
 2.130 
 
 0.532 
 
 1.881 
 
 1.133 
 
 0.883 
 
 62 
 
 29 
 
 0.485 
 
 2.063 
 
 0.554 
 
 1.804 
 
 1.143 
 
 0.875 
 
 61 
 
 30" 
 
 0.500 
 
 2.000 
 
 0.577 
 
 1.732 
 
 1.155 
 
 0.866 
 
 60° 
 
 31 
 
 0.515 
 
 1.942 
 
 0.601 
 
 1.664 
 
 1.167 
 
 0.857 
 
 59 
 
 32 
 
 0.530 
 
 1.8S7 
 
 0.625 
 
 1.600 
 
 1.179 
 
 0.848 
 
 58 
 
 33 
 
 0.545 
 
 1.836 
 
 0.649 
 
 1.540 
 
 1.192 
 
 0.839 
 
 57 
 
 34 
 
 0.559 
 
 1.788 
 
 0.675 
 
 1.483 
 
 1.206 
 
 0.829 
 
 56 
 
 35° 
 
 0.574 
 
 1.743 
 
 0.700 
 
 1.428 
 
 1.221 
 
 0.819 
 
 55° 
 
 36.. 
 '37 ;^ 
 S8 
 
 0.588 
 
 1.701 
 
 1 £.ilC%- ■■■ 
 
 0.727 
 
 1.376 
 
 -1.327 
 
 1.280 
 
 1.236 
 1.252 
 1.269 
 
 0.809 
 0.799 
 0.788 
 
 54 
 53 
 52 
 
 0.602 
 0.616 
 
 1.662 
 1.624 
 
 0.754- 
 0.781 
 
 39 
 
 0.629 
 
 1.589 
 
 0.810 
 
 1.235 
 
 1.287 
 
 0.777 
 
 51 
 
 40° 
 
 0.643 
 
 1.556 
 
 0.839 
 
 1.192 
 
 1.305 
 
 0.766 
 
 50° ' 
 
 41 
 
 0.656 
 
 1.524 
 
 0.869 
 
 1.150 
 
 1.325 
 
 0.755 
 
 49 
 
 42 
 
 0.669 
 
 1.494 
 
 0.900 
 
 1.111 
 
 1.346 
 
 0.743 
 
 48 
 
 43 
 
 0.682 
 
 1.466 
 
 0.933 
 
 1.072 
 
 1.367 
 
 0.731 
 
 47 
 
 44 
 
 0.695 
 
 1.440 
 
 0.966 
 
 1.036 
 
 1.390 
 
 0.719 
 
 46 
 
 45° 
 
 0.707 
 
 1.414 
 
 1.000 
 
 1.000 
 
 1.414 
 
 0.707 
 
 45° 
 
 
 Cos. 
 
 Sec. 
 
 Ctn. 
 
 Tan. 
 
 Csc. 
 
 Sin. 
 
 Angle. 
 
142 
 
 TABLES. 
 Logarithms. 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P.P. 
 
 1.2- 3. 4- 5 
 
 lO 
 
 0000 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 4- 8-12.17.21 
 
 11 
 
 0414 
 
 0453 
 
 0492 
 
 0531 
 
 0569 
 
 0607 
 
 0645 
 
 0682 
 
 0719 
 
 0755 
 
 4 8.11.10-19 
 
 12 
 
 0792 
 
 0828 
 
 0864 
 
 0899 
 
 0934 
 
 0969 
 
 1004 
 
 1038 
 
 1072 
 
 1106 
 
 3- 7-10. 14-17 
 
 13 
 
 1139 
 
 1173 
 
 1206 
 
 1239 
 
 1271 
 
 1303 
 
 1335 
 
 1367 
 
 1399 
 
 1430 
 
 3- 610.13.16 
 
 14 
 15 
 
 1461 
 
 1492 
 
 1523 
 
 1553 
 
 1584 
 
 1614 
 
 1044 
 
 1673 
 
 1703 
 
 1732 
 
 3. 6. 9.12.15 
 
 1761 
 
 1790 
 
 1818 
 
 1847 
 
 1875 
 
 1903 
 
 1931 
 
 1959 
 
 1987 
 
 2014 
 
 3. 6. 8-11.14 
 
 16 
 
 2041 
 
 2068 
 
 2095 
 
 2122 
 
 2148 
 
 2175 
 
 2201 
 
 2227 
 
 2253 
 
 2279 
 
 3. 5. 81113 
 
 17 
 
 2304 
 
 2330 
 
 2355 
 
 2380 
 
 2405 
 
 2430 
 
 2455 
 
 2480 
 
 2504 
 
 2529 
 
 2. 5- 7-10-12 
 
 18 
 
 2553 
 
 2577 
 
 2601 
 
 2625 
 
 2648 
 
 2672 
 
 2695 
 
 2718 
 
 2742 
 
 2765 
 
 2- 5- 7. 9-12 
 
 19 
 20 
 
 2788 
 
 2810 
 
 2833 
 
 2856 
 
 2878 
 
 2900 
 
 2923 
 
 2945 
 
 2967 
 
 2989 
 
 2. 4. 7. 911 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 2- 4- 6- 8 11 
 
 21 
 
 3222 
 
 3243 
 
 3263 
 
 3284 
 
 3304 
 
 3324 
 
 3345 
 
 3365 
 
 3385 
 
 3404 
 
 2. 4. 6. 810 
 
 22 
 
 3424 
 
 3444 
 
 3464 
 
 3483 
 
 3502 
 
 3522 
 
 3541 
 
 3560 
 
 3579 
 
 3598 
 
 2- 4. 6- 8.10 
 
 23 
 
 3617 
 
 3636 
 
 3655 
 
 3674 
 
 3692 
 
 3711 
 
 3729 
 
 3747 
 
 3766 
 
 3784 
 
 2. 4. 5- 7- 9 
 
 24 
 25 
 
 3802 
 
 3820 
 
 3838 
 
 3856 
 
 3874 
 
 3892 
 
 3909 
 
 3927 
 
 3945 
 
 3962 
 
 2- 4. 5- 7. 9 
 
 3979 
 
 3997 
 
 4014 
 
 4031 
 
 4048 
 
 4065 
 
 4082 
 
 4099 
 
 4110 
 
 4133 
 
 2 3. 5. 7. 9 
 
 26 
 
 4150 
 
 4166 
 
 4183 
 
 4200 
 
 4216 
 
 4232 
 
 4249 
 
 4265 
 
 4281 
 
 4298 
 
 2. 3. 5. 7. 8 
 
 27 
 
 4314 
 
 4330 
 
 4346 
 
 4362 
 
 4378 
 
 4393 
 
 4409 
 
 4425 
 
 4440 
 
 4456 
 
 2. 3. 5. 6- 8 
 
 28 
 
 4472 
 
 4487 
 
 4502 
 
 4518 
 
 4533 
 
 4548 
 
 4564 
 
 4579 
 
 4594 
 
 4609 
 
 2- 3. 5. 6. 8 
 
 29 
 30 
 
 4624 
 
 4639 
 
 4654 
 
 4669 
 
 4683 
 
 4698 
 
 4713 
 
 4728 
 
 4742 
 
 4757 
 
 1. 3- 4. 6. 7 
 
 4771 
 
 4786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 1- 3. 4. 6. 7 
 
 31 
 
 4914 
 
 4928 
 
 4942 
 
 4955 
 
 4969 
 
 4983 
 
 4997 
 
 5011 
 
 5024 
 
 5038 
 
 1- 3- 4. 6- 7 
 
 32 
 
 5051 
 
 5065 
 
 5079 
 
 5092 
 
 5105 
 
 5119 
 
 5132 
 
 5145 
 
 5159 
 
 5172 
 
 1. 3. 4- 5- 7 
 
 33 
 
 5185 
 
 5198 
 
 5211 
 
 5224 
 
 5237 
 
 5250 
 
 5263 
 
 5276 
 
 5289 
 
 5302 
 
 1- 3. 4- 5. 6 
 
 34 
 35 
 
 5315 
 
 5328 
 
 5340 
 
 5353 
 
 5366 
 
 5378 
 
 5391 
 
 5403 
 
 5416 
 
 5428 
 
 1. 3- 4- 5- 6 
 
 5441 
 
 5453 
 
 5465 
 
 5478 
 
 5490 
 
 5502 
 
 5514 
 
 5527 
 
 5539 
 
 5551 
 
 1- 2- 4- 5- 6 
 
 36 
 
 5563 
 
 5575 
 
 5587 
 
 5599 
 
 5611 
 
 5623 
 
 5635 
 
 5647 
 
 5658 
 
 5670 
 
 1- 2- 4. 5. 6 
 
 37 
 
 5682 
 
 5694 
 
 5705 
 
 5717 
 
 5729 
 
 5740 
 
 5752 
 
 5763 
 
 5775 
 
 5786 
 
 1-2. 3- 5. 6 
 
 38 
 
 5798 
 
 5809 
 
 5821 
 
 5832 
 
 58i3 
 
 5855 
 
 5866 
 
 5877 
 
 5888 
 
 5899 
 
 1- 2- 3. 5. 6 
 
 39 
 40 
 
 5911 
 
 5922 
 
 5933 
 
 5944 
 
 5955 
 
 5966 
 
 5977 
 
 5988 
 
 5999 
 
 6010 
 
 1- 2- 3. 4- 6 
 
 6021 
 
 6031 
 
 6042 
 
 6053 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 1- 2- 3. 4. 5 
 
 41 
 
 6128 
 
 6138 
 
 6149 
 
 6160 
 
 6170 
 
 6180 
 
 6191 
 
 6201 
 
 6212 
 
 6222 
 
 1- 2. 3. 4. 5 
 
 42 
 
 6232 
 
 6243 
 
 6253 
 
 6263 
 
 6274 
 
 6284 
 
 6294 
 
 6304 
 
 6314 
 
 6325 
 
 1- 2- 3- 4. 5 
 
 43 
 
 6335 
 
 6345 
 
 6355 
 
 6365 
 
 6375 
 
 6385 
 
 6395 
 
 6405 
 
 6415 
 
 6425 
 
 1- 2. 3. 4- 5 
 
 44 
 45 
 
 6435 
 
 6444 
 
 6454 
 
 6464 
 
 6474 
 
 6484 
 
 6493 
 
 6503 
 
 6513 
 
 6522 
 
 1. 2. 3- 4- 5 
 
 6532 
 
 6542 
 
 6551 
 
 65G1 
 
 6571 
 
 6580 
 
 6590 
 
 6599 
 
 6809 
 
 6618 
 
 1. 2- 3. 4. 5 
 
 46 
 
 6628 
 
 6637 
 
 6646 
 
 6656 
 
 6665 
 
 6675 
 
 6684 
 
 6693 
 
 6702 
 
 6712 
 
 1. 2- 3- 4- 5 
 
 47 
 
 6721 
 
 6730 
 
 6739 
 
 6749 
 
 6758 
 
 6767 
 
 6776 
 
 6785 
 
 6794 
 
 6803 
 
 1. 2. 3. 4. 5 
 
 48 
 
 6812 
 
 6821 
 
 6830 
 
 6839 
 
 6848 
 
 6857 
 
 6366 
 
 6875 
 
 6884 
 
 6893 
 
 1. 2. 3 4. 4 
 
 49 
 50 
 
 6902 
 
 6911 
 
 6920 
 
 6928 
 
 6937 
 
 6946 
 
 6955 
 
 6964 
 
 6972 
 
 6981 
 
 1- 2- 3- 4- 4 
 
 6990 
 
 6998 
 
 7007 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 1- 2. 3. 3. 4 
 
 51 
 
 7076 
 
 7084 
 
 7093 
 
 7101 
 
 7110 
 
 7118 
 
 7126 
 
 7135 
 
 7143 
 
 7152 
 
 1- 2- 3. 3. 4 
 
 52 
 
 7160 
 
 7168 
 
 7177 
 
 7185 
 
 7193 
 
 7202 
 
 7210 
 
 7218 
 
 7226 
 
 7235 
 
 1- 2- 2- 3. 4 
 
 53 
 
 7243 
 
 7251 
 
 7259 
 
 7267 
 
 7275 
 
 7284 
 
 7292 
 
 7300 
 
 7308 
 
 7316 
 
 1- 2. 2- 3. 4 
 
 54 
 
 7324 
 
 7332 
 
 7340 
 
 7348 
 
 7356 
 
 7364 
 
 7372 
 
 7380 
 
 7388 
 
 7396 
 
 1. 2- 2. 3. 4 
 
 NoTK. — This page and the three that follow it are taken from the Mathematical 
 Tables of Prof. J. M. Peirce, published by ^Messrs. Ginn & Co. 
 
TABLES. 
 Logarithms. 
 
 143 
 
 [ 
 
 N 
 
 
 
 12 3 
 
 4 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 P P. 
 
 
 
 
 
 
 
 
 
 1. ^ 
 
 • 3- 4. 5 
 
 55 
 
 7404 
 
 7412 7419 7427 
 
 7435 
 
 7443 
 
 7451 
 
 7459 
 
 7466 
 
 7474 
 
 J. 2 
 
 . 2. 3- 4 
 
 56 
 
 7482 
 
 7490 7497 7505 
 
 7513 
 
 7520 
 
 7528 
 
 7536 
 
 7543 
 
 7551 
 
 1-2 
 
 .23-4 
 
 57 
 
 7559 
 
 7566 7574 7582 
 
 7589 
 
 7597 
 
 7604 
 
 7612 
 
 7619 
 
 7627 
 
 1. 2 
 
 • 2. 3. 4 
 
 58 
 
 7634 
 
 7642 7649 7657 
 
 7664 
 
 7672 
 
 7679 
 
 7686 
 
 7694 
 
 7701 
 
 
 2-3. 4 
 
 59 
 60 
 
 7709 
 
 7716 7723 7731 
 
 7738 
 
 7745 
 
 7752 
 
 7760 
 
 7767 
 
 7774 
 
 
 2. 3 4 
 
 7782 
 
 7789 7796 7803 
 
 7810 
 
 7818 
 
 7825 
 
 7832 
 
 7839 
 
 7846 
 
 
 2-3.4 
 
 61 
 
 7853 
 
 7860 7868 7875 
 
 7882 
 
 7889 
 
 7896 
 
 7903 
 
 7910 
 
 7917 
 
 
 2.3-4 
 
 62 
 
 7924 
 
 7931 7938 7945 
 
 7952 
 
 7959 
 
 7966 
 
 7973 
 
 7980 
 
 7987 
 
 
 2.3.3 
 
 63 
 
 7993 
 
 8000 8007 8014 
 
 8021 
 
 8028 
 
 8035 
 
 8041 
 
 8048 
 
 8055 
 
 
 2.3- 3 
 
 64 
 65 
 
 8062 
 
 8069 8075 8082 
 
 8089 
 
 8096 
 
 8102 
 
 8109 
 
 8116 
 
 8122 
 
 
 2.3.3 
 
 8129 
 
 8136 8142 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 
 2-3.3 
 
 66 
 
 8195 
 
 8202 8200 8215 
 
 8222 
 
 8228 
 
 8235 
 
 8241 
 
 8248 
 
 8254 
 
 
 2.3.3 
 
 67 
 
 8261 
 
 8267 8274 8280 
 
 8287 
 
 8293 
 
 8299 
 
 8306 
 
 8312 
 
 8319 
 
 
 2.3.3 
 
 68 
 
 8325 
 
 8331 8338 8344 
 
 8351 
 
 8357 
 
 8363 
 
 8370 
 
 8376 
 
 8382 
 
 
 2.3. 3 
 
 69 
 70 
 
 8388 
 
 8395 8401 8407 
 
 8414 
 
 8420 
 
 8426 
 
 8432 
 
 8439 
 
 8445 
 
 
 2. 3.3 
 
 8451 
 
 8457 8463 8470 
 
 8476 
 
 8482 
 
 8488 
 
 8494 
 
 8500 
 
 8506 
 
 
 2.2.3 
 
 71 
 
 8513 
 
 8519 8525 8531 
 
 8537 
 
 8543 
 
 8549 
 
 8555 
 
 8561 
 
 8567 
 
 
 2-2.3 
 
 72 
 
 8573 
 
 8579 8585 8591 
 
 8597 
 
 8603 
 
 8609 
 
 8615 
 
 8621 
 
 8627 
 
 
 2.2.3 
 
 73 
 
 8633 
 
 8639 8645 8651 
 
 8657 
 
 8663 
 
 8669 
 
 8675 
 
 8681 
 
 8686 
 
 
 2. 2-3 
 
 74 
 75 
 
 8692 
 
 8698 8704 8710 
 
 8716 
 
 8722 
 
 8727 
 
 8733 
 
 8739 
 
 8745 
 
 
 2-2.3 
 
 8751 
 
 8756 8762 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 
 2.2.3 
 
 76 
 
 8808 
 
 8814 8820 8825 
 
 8831 
 
 8837 
 
 8842 
 
 8848 
 
 8854 
 
 8859 
 
 
 2-2-3 
 
 77 
 
 8865 
 
 8871 8876 8882 
 
 8887 
 
 8893 
 
 8899 
 
 8904 
 
 8910 
 
 8915 
 
 
 2.2.3 
 
 78 
 
 8921 
 
 8927 8932 8938 
 
 8943 
 
 8949 
 
 8954 
 
 8960 
 
 8965 
 
 8971 
 
 
 2.2.3 
 
 79 
 80 
 
 8976 
 
 8982 8987 8993 
 
 8998 
 
 9004 
 
 9009 
 
 9015 
 
 9020 
 
 9025 
 
 
 2- 2. 3 
 
 9031 
 
 9036 9042 9047 
 
 9053 
 
 9058 
 
 9063 
 
 9069 
 
 9074 
 
 9079 
 
 
 2-2.3 
 
 81 
 
 9085 
 
 9090 9096 9101 
 
 9106 
 
 9112 
 
 9117 
 
 9122 
 
 9128 
 
 9133 
 
 
 2.2.3 
 
 82 
 
 9138 
 
 9143 9149 9154 
 
 9159 
 
 9165 
 
 9170 
 
 9175 
 
 9180 
 
 9186 
 
 
 2.2-3 
 
 83 
 
 9191 
 
 9196 9201 9206 
 
 9212 
 
 9217 
 
 9222 
 
 9227 
 
 9232 
 
 9238 
 
 
 2.2-3 
 
 84 
 85 
 
 9243 
 
 9248 9253 9258 
 
 9263 
 
 9269 
 
 9274 
 
 9279 
 
 9284 
 
 9289 
 
 
 2.2.3 
 
 9294 
 
 9299 9304 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 
 2 2.3 
 
 86 
 
 9345 
 
 9350 9355 9360 
 
 9365 
 
 9370 
 
 9375 
 
 9380 
 
 9385 
 
 9390 
 
 
 2-2.3 
 
 87 
 
 9395 
 
 9400 9405 9410 
 
 9415 
 
 9420 
 
 9425 
 
 9430 
 
 9435 
 
 9440 
 
 0. 1 
 
 1.2.2 
 
 88 
 
 9445 
 
 9450 9455 9460 
 
 9465 
 
 9469 
 
 9474 
 
 9479 
 
 9484 
 
 9489 
 
 0.1 
 
 1- 2-2 
 
 89 
 90 
 
 9494 
 
 9499 9504 9509 
 
 9513 
 
 9518 
 
 9523 
 
 9528 
 
 9533 
 
 9538 
 
 0. 1 
 
 1.2.2 
 
 9542 
 
 9547 9552 9557 
 
 9562 
 
 9566 
 
 9571 
 
 9576 
 
 9581 
 
 9586 
 
 0- 1 
 
 1.2.2 
 
 91 
 
 9590 
 
 9595 9600 9605 
 
 9609 
 
 9614 
 
 9619 
 
 9624 
 
 9628 
 
 9633 
 
 0. 1 
 
 1.2.2 
 
 92 
 
 9638 
 
 9643 9647 9652 
 
 9657 
 
 9661 
 
 9666 
 
 9671 
 
 9675 
 
 9680 
 
 0- 1 
 
 1.2-2 
 
 93 
 
 9685 
 
 9689 9694 9699 
 
 9703 
 
 9708 
 
 9713 
 
 9717 
 
 9722 
 
 9727 
 
 0- 1 
 
 1.2.2 
 
 94 
 95 
 
 9731 
 
 9736 9741 9745 
 
 9750 
 
 9754 
 
 9759 
 
 9763 
 
 9768 
 
 9773 
 
 1 
 
 1. 2- 2 
 
 9777 
 
 9782 9786 9791 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 0. 1 
 
 1.2.2 
 
 96 
 
 9823 
 
 9827 9832 9836 
 
 9841 
 
 9845 
 
 9850 
 
 9854 
 
 9859 
 
 9863 
 
 0. 1 
 
 1.2.2 
 
 97 
 
 9868 
 
 9872 9877 9881 
 
 9886 
 
 9890 
 
 9894 
 
 9899 
 
 9903 
 
 9908 
 
 0- 1 
 
 1.2.2 
 
 98 
 
 9912 
 
 9917 9921 9926 
 
 9930 
 
 9934 
 
 9939 
 
 9943 
 
 9948 
 
 9952 
 
 0- 1 
 
 . 1. 2. 2 
 
 99 
 
 9956 
 
 9961 9965 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 9991 
 
 9996 
 
 0- 1 
 
 1.2-2 
 
 log !r= 0.49715- 
 
 log e = 0.43429 - 
 
144 
 
 TABLES. 
 
 Logarithms. 
 
 N 
 
 
 
 T 
 
 2 
 
 3 4 5 
 
 6 
 
 7 
 
 8 
 
 9 10 
 
 100 
 
 0000 
 
 0004 
 
 0009 
 
 0013 0017 
 
 0022 
 
 0026 
 
 0030 
 
 0035 
 
 0039 
 
 0043 
 
 101 
 
 0043 
 
 0043 
 
 0052 
 
 0056 0060 
 
 0065 
 
 0069 
 
 0073 
 
 0077 
 
 0082 
 
 0086 
 
 102 
 
 0086 
 
 0090 
 
 0095 
 
 0099 0103 
 
 0107 
 
 0111 
 
 0116 
 
 0120 
 
 0124 
 
 0128 
 
 103 
 
 0128 
 
 0133 
 
 0137 
 
 0141 0145 
 
 0149 
 
 0154 
 
 0158 
 
 0162 
 
 0166 
 
 0170 
 
 104 
 105 
 
 0170 
 
 0175 
 
 0179 
 
 0183 0187 
 
 0191 
 
 0195 
 
 0199 
 
 0204 
 
 0208 
 
 0212 
 
 0212 
 
 0216 
 
 0220 
 
 0224 0228 
 
 0233 
 
 0237 
 
 0241 
 
 0245 
 
 0249 
 
 0253 
 
 106 
 
 0253 
 
 0257 
 
 0261 
 
 0265 0269 
 
 0273 
 
 0278 
 
 0282 
 
 0286 
 
 0290 
 
 0294 
 
 107 
 
 0294 
 
 0298 
 
 0302 
 
 0306 0310 
 
 0314 
 
 0318 
 
 0322 
 
 0326 
 
 0330 
 
 0334 
 
 108 
 
 0334 
 
 0338 
 
 0342 
 
 0346 0350 
 
 0354 
 
 0358 
 
 0362 
 
 0366 
 
 0370 
 
 0374 
 
 109 
 110 
 
 0374 
 
 0378 
 
 0382 
 
 0386 0390 
 
 0394 
 
 0398 
 
 0402 
 
 0406 
 
 0410 
 
 0414 
 
 0414 
 
 0418 
 
 0422 
 
 0426 0430 
 
 0434 
 
 0438 
 
 0441 
 
 0445 
 
 0449 
 
 0453 
 
 111 
 
 0453 
 
 0457 
 
 0461 
 
 0465 0469 
 
 0473 
 
 0477 
 
 0481 
 
 0484 
 
 0488 
 
 0492 
 
 112 
 
 0492 
 
 0496 
 
 0500 
 
 0504 0508 
 
 0512 
 
 0515 
 
 0519 
 
 0523 
 
 0527 
 
 0531 
 
 113 
 
 0531 
 
 0535 
 
 0538 
 
 0542 0546 
 
 0550 
 
 0554 
 
 0558 
 
 0561 
 
 0565 
 
 0569 
 
 114 
 115 
 
 0569 
 
 0573 
 
 0577 
 
 0580 0584 
 
 0588 
 
 0592 
 
 0596 
 
 0599 
 
 0603 
 
 0607 
 
 0607 
 
 0611 
 
 0615 
 
 0618 0622 
 
 0626 
 
 0630 
 
 0633 
 
 0637 
 
 0641 
 
 0645 
 
 116 
 
 0645 
 
 0648 
 
 0652 
 
 0656 0660 
 
 0663 
 
 0667 
 
 0671 
 
 0674 
 
 0678 
 
 0682 
 
 117 
 
 0682 
 
 0686 
 
 0689 
 
 0693 0697 
 
 0700 
 
 0704 
 
 0708 
 
 0711 
 
 0715 
 
 0719 
 
 118 
 
 0719 
 
 0722 
 
 0726 
 
 0730 0734 
 
 0737 
 
 0741 
 
 0745 
 
 0748 
 
 0752 1 0755 
 
 119 
 120 
 
 0755 
 
 0759 
 
 0763 
 
 0766 0770 
 
 0774 
 
 0777 
 
 0781 
 
 0785 
 
 0788 \ 0792 
 
 0792 
 
 0795 
 
 0799 
 
 0803 0806 
 
 0810 
 
 0813 
 
 0817 
 
 0821 
 
 0824 
 
 0828 
 
 121 
 
 0828 
 
 0831 
 
 0835 
 
 0839 0842 ' 0846 
 
 0849 
 
 0853 
 
 0856 
 
 0860 
 
 0864 
 
 122 
 
 0864 
 
 0867 
 
 0871 
 
 0874 0878 / 0881 
 
 0885 
 
 0888 
 
 0892 
 
 0896 
 
 0899 
 
 123 
 
 0899 
 
 0903 
 
 0906 
 
 0910 0913 
 
 0917 
 
 0920 
 
 0924 
 
 0927 
 
 0931 
 
 0934 
 
 124 
 125 
 
 0934 
 
 0938 
 
 0941 
 
 0945 0948 
 
 0952 
 
 0955 
 
 0959 
 
 0962 
 
 0966 
 
 0969 
 
 0969 
 
 0973 
 
 0976 
 
 0980 0983 
 
 0986 
 
 0990 
 
 0993 
 
 0997 
 
 1000 
 
 1004 
 
 126 
 
 1004 
 
 1007 
 
 1011 
 
 1014 1017 
 
 1021 
 
 1024 
 
 1028 
 
 1031 
 
 1035 
 
 1038 
 
 127 
 
 1038 
 
 1041 
 
 1045 
 
 1048 1052 
 
 1055 
 
 1059 
 
 1062 
 
 1065 
 
 1069 
 
 1072 
 
 128 
 
 1072 
 
 1075 
 
 1079 
 
 1082 1086 
 
 1089 
 
 1092 
 
 1096 
 
 1099 
 
 1103 
 
 1106 
 
 129 
 130 
 
 1106 
 
 1109 
 
 1113 
 
 1116 1119 
 
 1123 
 
 1126 
 
 1129 
 
 1133 
 
 1136 
 
 1139 
 
 1139 
 
 1143 
 
 1146 
 
 1149 1153 
 
 1156 
 
 1159 
 
 1163 
 
 1166 
 
 1169 
 
 1173 
 
 131 
 
 1173 
 
 1176 
 
 1179 
 
 1183 1186 
 
 1189 
 
 1193 
 
 1196 
 
 1199 
 
 1202 
 
 1206 
 
 132 
 
 1208 
 
 1209 
 
 1212 
 
 1216 1219 
 
 1222 
 
 1225 
 
 1229 
 
 1232 
 
 1235 
 
 1239 
 
 133 
 
 1239 
 
 1242 
 
 1245 
 
 1248 1252 
 
 1255 
 
 1258 
 
 1261 
 
 1265 
 
 1268 
 
 1271 
 
 134 
 135 
 
 1271 
 
 1274 
 
 1278 
 
 1281 1284 
 
 1287 
 
 1290 
 
 1294 
 
 1297 
 
 1300 
 
 1303 
 
 1303 
 
 1307 
 
 1310 
 
 1313 1316 
 
 1319 
 
 1323 
 
 1326 
 
 1329 
 
 1332 
 
 1335 
 
 136 
 
 1335 
 
 1339 
 
 1342 
 
 1345 1348 
 
 1351 
 
 1355 
 
 1358 
 
 1361 
 
 1364 
 
 1367 
 
 137 
 
 1367 
 
 1370 
 
 1374 
 
 1377 1380 
 
 1383 
 
 1386 
 
 1389 
 
 1392 
 
 1396 
 
 1399 
 
 138 
 
 1399 
 
 1402 
 
 1405 
 
 1408 1411 
 
 1414 
 
 1418 
 
 1421 
 
 1424 
 
 1427 
 
 1430 
 
 139 
 140 
 
 1430 
 
 1433 
 
 1436 
 
 1440 1443 
 
 1446 
 
 1449 
 
 1452 
 
 1455 
 
 1458 
 
 1461 
 
 1461 
 
 1464 
 
 1467 
 
 1471 1474 
 
 1477 
 
 1480 
 
 1483 
 
 1486 
 
 1489 
 
 1492 
 
 141 
 
 1492 
 
 1495 
 
 1498 
 
 1501 1504 
 
 1508 
 
 1511 
 
 1514 
 
 1517 
 
 1520 
 
 1523 
 
 142 
 
 1523 
 
 1526 
 
 1529 
 
 1532 1535 
 
 1538 
 
 1541 
 
 1544 
 
 1547 
 
 1550 
 
 1553 
 
 143 
 
 1553 
 
 1556 
 
 1559 
 
 1562 1565 
 
 1569 
 
 1572 
 
 1575 
 
 1578 
 
 1581 
 
 1584 
 
 144 
 145 
 
 1584 
 
 1587 
 
 1590 
 
 1593 1596 
 
 1599 
 
 1602 
 
 1605 
 
 1608 
 
 1611 
 
 1614 
 
 1614 
 
 1617 
 
 1620 
 
 1623 1626 
 
 1629 
 
 1632 
 
 1635 
 
 1638 
 
 1641 
 
 1644 
 
 146 
 
 1644 
 
 1647 
 
 1649 
 
 1652 1655 
 
 1658 
 
 1661 
 
 1664 
 
 1667 
 
 1670 
 
 1673 
 
 147 
 
 1673 
 
 1676 
 
 1679 
 
 1682 1685 
 
 1688 
 
 1691 
 
 1694 
 
 1697 
 
 1700 
 
 1703 
 
 148 
 
 1703 
 
 1706 
 
 1708 
 
 1711 1714 
 
 1717 
 
 1720 
 
 1723 
 
 1726 
 
 1729 
 
 1732 
 
 149 
 
 1732 
 
 1735 
 
 1738 
 
 1741 1744 
 
 1746 
 
 1749 
 
 1752 
 
 1755 
 
 1758 
 
 1761 
 
TABLES. 
 
 Logarithms. 
 
 145 
 
 N 
 
 
 
 1 2 
 
 3 
 
 4 5 
 
 6 
 
 7 8 
 
 9 
 
 10 
 
 150 
 
 1761 
 
 1764 1767 
 
 1770 
 
 1772 
 
 1775 
 
 1778 
 
 1781 1784 
 
 1787 
 
 1790 
 
 151 
 
 1790 
 
 1793 1796 
 
 1798 
 
 1801 
 
 1804 
 
 1807 
 
 1810 1813 
 
 1816 
 
 1818 
 
 152 
 
 1818 
 
 1821 1824 
 
 1827 
 
 1830 
 
 1833 
 
 1836 
 
 1838 1841 
 
 1844 
 
 1847 
 
 153 
 
 1847 
 
 1850 1853 
 
 1855 
 
 1858 
 
 1861 
 
 1864 
 
 1867 1870 
 
 1872 
 
 1875 
 
 154 
 155 
 
 1875 
 
 1878 1881 
 
 1884 
 
 1886 
 
 1889 
 
 1892 
 
 1895 1898 
 
 1901 
 
 1903 
 
 1903 
 
 1906 1909 
 
 1912 
 
 1915 
 
 1917 
 
 1920 
 
 1923 1926 
 
 1928 
 
 1931 
 
 156 
 
 1931 
 
 1934 1937 
 
 1940 
 
 1942 
 
 1945 
 
 1948 
 
 1951 1953 
 
 1956 
 
 1959 
 
 157 
 
 1959 
 
 1962 1965 
 
 1967 
 
 1970 
 
 1973 
 
 1976 
 
 1978 1981 
 
 1984 
 
 1987 
 
 158 
 
 1987 
 
 1989 1992 
 
 1995 
 
 1998 
 
 2000 
 
 2003 
 
 2006 2009 
 
 2011 
 
 2014 
 
 159 
 160 
 
 2014 
 
 2017 2019 
 
 2022 
 
 2025 
 
 2028 
 
 2030 
 
 2033 2036 
 
 2038 
 
 2041 
 
 2041 
 
 2044 2047 
 
 2049 
 
 2052 
 
 2055 
 
 2057 
 
 2060 2063 
 
 2066 
 
 2068 
 
 161 
 
 2068 
 
 2071 2074 
 
 2076 
 
 2079 
 
 2082 
 
 2084 
 
 2087 2090 
 
 2092 
 
 2095 
 
 162 
 
 2095 
 
 2098 2101 
 
 2103 
 
 2106 
 
 2109 
 
 2111 
 
 2114 2117 
 
 2119 
 
 2122 
 
 163 
 
 2122 
 
 2125 2127 
 
 2130 
 
 2133 
 
 2135 
 
 2138 
 
 2140 2143 
 
 2146 
 
 2148 
 
 164 
 165 
 
 2148 
 
 2151 2154 
 
 2156 
 
 2159 
 
 2162 
 
 2164 
 
 2167 2170 
 
 2172 
 
 2175 
 
 2175 
 
 2177 2180 
 
 2183 
 
 2185 
 
 2188 
 
 2391 
 
 2193 2196 
 
 2198 
 
 2201 
 
 166 
 
 2201 
 
 2204 2206 
 
 2209 
 
 2212 
 
 2214 
 
 2217 
 
 2219 2222 
 
 2225 
 
 2227 
 
 167 
 
 2227 
 
 2230 2232 
 
 2235 
 
 2238 
 
 2240 
 
 2243 
 
 2245 2248 
 
 2251 
 
 2253 
 
 168 
 
 2253 
 
 2256 2258 
 
 2261 
 
 2263 
 
 2266 
 
 2269 
 
 2271 2274 
 
 2276 
 
 2279 
 
 169 
 170 
 
 2279 
 
 2281 2284 
 
 2287 
 
 2289 
 2315 
 
 2292 
 
 2294 
 
 2297 2299 
 
 2302 
 
 2304 
 2330 
 
 2304 
 
 2307 2310 
 
 2312 
 
 2317 
 
 2320 
 
 2322 2325 
 
 2327 
 
 171 
 
 2330 
 
 2333 2335 
 
 2338 
 
 2340 
 
 2343 
 
 2345 
 
 2348 2350 
 
 2353 
 
 2355 
 
 172 
 
 2355 
 
 2358 2360 
 
 2363 
 
 2305 
 
 2368 
 
 2370 
 
 2373 2375 
 
 2378 
 
 2380 
 
 173 
 
 2380 
 
 2383 2385 
 
 2388 
 
 2390 
 
 2393 
 
 2395 
 
 2398 2400 
 
 2403 
 
 2405 
 
 174 
 175 
 
 2405 
 
 2408 2410 
 
 2413 
 
 2415 
 
 2418 
 
 2420 
 
 2423 2425 
 
 2428 
 
 2430 
 
 2430 
 
 2433 2435 
 
 2438 
 
 2440 
 
 2443 
 
 2445 
 
 2448 2450 
 
 2453 
 
 2455 
 
 176 
 
 2455 
 
 2458 2460 
 
 2463 
 
 2465 
 
 2467 
 
 2470 
 
 2472 2475 
 
 2477 
 
 2480 
 
 177 
 
 2480 
 
 2482 2485 
 
 2487 
 
 2490 
 
 2492 
 
 2494 
 
 2497 2499 
 
 2502 
 
 2504 
 
 178 
 
 2504 
 
 2507 2509 
 
 2512 
 
 2514 
 
 2516 
 
 2519 
 
 2521 2524 
 
 2526 
 
 2529 
 
 179 
 180 
 
 2529 
 
 2531 2533 
 
 2536 
 
 2538 2541 
 
 2543 
 
 2545 2548 
 
 2550 
 
 2553 
 
 2553 
 
 2555 2558 
 
 2560 
 
 2562 ' 
 
 2565 
 
 2567 
 
 2570 2572 
 
 2574 
 
 2577 
 
 181 
 
 2577 
 
 2579 2582 
 
 2584 
 
 2586 
 
 2589 
 
 2591 
 
 2594 2596 
 
 2598 
 
 2601 
 
 182 
 
 2601 
 
 2603 2605 
 
 2608 
 
 2610 
 
 2613 
 
 2615 
 
 2617 2620 
 
 2622 
 
 2625 
 
 183 
 
 2625 
 
 2627 2629 
 
 2632 
 
 2634 
 
 2636 
 
 2639 
 
 2641 2643 
 
 2646 
 
 2648 
 
 184 
 185 
 
 2648 
 
 2651 2653 2655 
 
 2658 
 
 2660 
 
 2662 
 
 2665 2667 
 
 2669 
 
 2672 
 
 2672 
 
 2674 2676 
 
 2679 
 
 2681 
 
 2083 
 
 2686 
 
 2688 2690 
 
 2693 
 
 2695 
 
 186 
 
 2695 
 
 2697 2700 
 
 2702 
 
 2704 
 
 2707 
 
 2709 
 
 2711 2714 
 
 2716 
 
 2718 
 
 187 
 
 2718 
 
 2721 2723 
 
 2725 
 
 2728 
 
 2730 
 
 2732 
 
 2735 2737 
 
 2739 
 
 2742 
 
 188 
 
 2742 
 
 2744 2746 
 
 2749 
 
 2751 
 
 2753 
 
 2755 
 
 2758 2760 
 
 2762 
 
 2765 
 
 189 
 190 
 
 2765 
 
 2767 2769 
 
 2772 
 
 2774 
 
 2776 
 
 2778 
 
 2781 2783 
 
 2785 
 
 2788 
 
 2788 
 
 2790 2792 
 
 2794 
 
 2797 
 
 2799 
 
 2801 
 
 2804 2806 
 
 2808 
 
 2810 
 
 191 
 
 2810 
 
 2813 2815 
 
 2817 
 
 2819 
 
 2822 
 
 2824 
 
 2826 2828 
 
 2831 
 
 2833 
 
 192 
 
 2833 
 
 2835 2838 
 
 2840 
 
 2842 
 
 2844 
 
 2847 
 
 2849 2851 
 
 2853 
 
 2856 
 
 193 
 
 2856 
 
 2858 2860 
 
 2862 
 
 2865 
 
 2867 
 
 2869 
 
 2871 2874 
 
 2876 
 
 2878 
 
 194 
 195 
 
 2878 
 
 2880 2882 
 
 2885 
 
 2887 
 
 2889 
 
 2891 
 
 2894 2896 
 
 2898 
 
 2900 
 
 2900 
 
 2903 2905 
 
 2907 
 
 2909 
 
 2911 
 
 2914 
 
 2916 2918 
 
 2920 
 
 2923 
 
 196 
 
 2923 
 
 2925 2927 
 
 2929 
 
 2931 
 
 2934 
 
 2936 
 
 2938 2940 
 
 2942 
 
 2945 
 
 197 
 
 2945 
 
 2947 2949 
 
 2951 
 
 2953 
 
 2956 
 
 2958 
 
 2960 2962 
 
 2964 
 
 2967 
 
 198 
 
 2967 
 
 2969 2971 
 
 2973 
 
 2975 
 
 2978 
 
 2980 
 
 2982 2984 
 
 2986 
 
 2989 
 
 199 
 
 2989 
 
 2991 2993 
 
 2995 
 
 2997 
 
 2999 
 
 3002 
 
 3004 3006 
 
 3008 
 
 3010 
 
146 
 
 TABLES. 
 
 Trigonometric Functions. 
 
 RADIANS. 
 
 DEGREES. 
 
 SINES. 
 
 COSINES. 
 
 TANGENTS. 
 
 COTANGENTS. 
 
 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. 
 
 Log. 
 
 
 
 0.0000 
 
 0°00' 
 
 .0000 CO 
 
 1.0000 0.0000 
 
 .0000 00 
 
 CO 
 
 CO 
 
 90° 00' 
 
 1.5708 
 
 0.0029 
 
 10 
 
 .0029 7.4637 
 
 1.0000 .0000 
 
 .0029 7.4637 
 
 343.77 2.5363 
 
 50 
 
 1.5679 
 
 0.0058 
 
 20 
 
 .0058 .7648 
 
 1.0000 .0000 
 
 .0058 .7648 
 
 171.89 
 
 .2352 
 
 40 
 
 1.5650 
 
 0.0087 
 
 30 
 
 .0087 .9408 
 
 1.0000 .0000 
 
 .0087 .9409 
 
 114.59 
 
 .0591 
 
 30 
 
 1.5621 
 
 0.0116 
 
 40 
 
 .0116 8.0658 
 
 .9999 .0000 
 
 .0116 8.0658 
 
 85.940 
 
 1.9342 
 
 20 
 
 1.5592 
 
 0.0145 
 
 50 
 
 .0145 .1627 
 
 .9999 .0000 
 
 .0145 .1627 
 
 68.750 
 
 .8373 
 
 10 
 
 1.5563 
 
 0.0175 
 
 POO' 
 
 .0175 8.2419 
 
 .9998 9.9999 
 
 .0175 8.2419 
 
 57.290 
 
 1.7581 
 
 89° 00' 
 
 1.5533 
 
 0.02(H 
 
 10 
 
 .0204 .3088 
 
 .9998 .9999 
 
 .0204 .3089 
 
 49.104 
 
 .6911 
 
 50 
 
 1.5504 
 
 0.0233 
 
 20 
 
 .0233 .3668 
 
 .9997 .9999 
 
 .0233 .3669 
 
 42.964 
 
 .6331 
 
 40 
 
 1.5475 
 
 0.0262 
 
 30 
 
 .0262 .4179 
 
 .9997 .9999 
 
 .0262 .4181 
 
 38.188 
 
 .5819 
 
 30 
 
 1.5446 
 
 0.0291 
 
 40 
 
 .0291 .4637 
 
 .9996 .9998 
 
 .0291 .4638 
 
 34.368 
 
 .5362 
 
 20 
 
 1.5417 
 
 0.0320 
 
 50 
 
 .0320 .5050 
 
 .9995 .9998 
 
 .0320 .5053 
 
 31.242 
 
 .4947 
 
 10 
 
 1.5388 
 
 0.0349 
 
 2° 00' 
 
 .0349 8.5428 
 
 .9994 9.9997 
 
 .0349 8.5431 
 
 28.636 
 
 1.4569 
 
 88° 00' 
 
 1.5359 
 
 0.0378 
 
 10 
 
 .0378 .5776 
 
 .9993 .9997 
 
 .0378 .5779 
 
 26.432 
 
 .4221 
 
 50 
 
 1.5330 
 
 0.0407 
 
 20 
 
 .0407 .6097 
 
 .9992 .9996 
 
 .0407 .6101 
 
 24.542 
 
 .3899 
 
 40 
 
 1.5301 
 
 0.0436 
 
 30 
 
 .0436 .6397 
 
 .9990 .9996 
 
 .0437 .6401 
 
 22.904 
 
 .3599 
 
 30 
 
 1.5272 
 
 0.0465 
 
 40 
 
 .0465 .6677 
 
 .9989 .9995 
 
 .0466 .6682 
 
 21.470 
 
 .3318 
 
 20 
 
 1.5243 
 
 0.0495 
 
 50 
 
 .0494 .6940 
 
 .9988 .9995 
 
 .0495 .6945 
 
 20.206 
 
 .3055 
 
 10 
 
 1.5213 
 
 0.0524 
 
 3° 00' 
 
 .0523 8.7188 
 
 .9986 9.9994 
 
 .0524 8.7194 
 
 19.081 
 
 1.2806 
 
 87° 00' 
 
 1.5184 
 
 0.0553 
 
 10 
 
 .0552 .7423 
 
 .9985 .9993 
 
 .0553 .7429 
 
 18.075 
 
 .2571 
 
 50 
 
 1.5155 
 
 0.0582 
 
 20 
 
 .0581 .7645 
 
 .9983 .9993 
 
 .0582 .7652 
 
 17.169 
 
 .2348 
 
 40 
 
 1.5126 
 
 0.0611 
 
 30 
 
 .0610 .7857 
 
 .9981 .9992 
 
 .0612 .7865 
 
 16.350 
 
 .2135 
 
 30 
 
 1.5097 
 
 0.0640 
 
 40 
 
 .0640 .8059 
 
 .9980 .9991 
 
 .0641 .8067 
 
 15.605 
 
 .1933 
 
 20 
 
 1.5068 
 
 0.0669 
 
 50 
 
 .0669 .8251 
 
 .9978 .9990 
 
 .0670 .8261 
 
 14.924 
 
 .1739 
 
 10 
 
 1.5039 
 
 0.0698 
 
 4° 00' 
 
 .0698 S.S436 
 
 .9976 9.9989 
 
 .0699 8.8446 
 
 14.301 
 
 1.1554 
 
 86° 00' 
 
 1.5010 
 
 0.0727 
 
 10 
 
 .0727 .8613 
 
 .9974 .9989 
 
 .0729 .8624 
 
 13.727 
 
 .1376 
 
 50 
 
 1.4981 
 
 0.0756 
 
 20 
 
 .0756 .8783 
 
 .9971 .9988 
 
 .0758 .8795 
 
 13.197 
 
 .1205 
 
 40 
 
 1.4952 
 
 0.0785 
 
 30 
 
 .0785 .8946 
 
 .9969 .9987 
 
 .0787 .8960 
 
 12.706 
 
 .1040 
 
 30 
 
 1.4923 
 
 0.0814 
 
 40 
 
 .0814 .9104 
 
 .9967 .9986 
 
 .0816 .9118 
 
 12.251 
 
 .0882 
 
 20 
 
 1.4893 
 
 0.0844 
 
 50 
 
 .0843 .9256 
 
 .9964 .9985 
 
 .0846 .9272 
 
 11.826 
 
 .0728 
 
 10 
 
 1.4864 
 
 0.0873 
 
 5°00' 
 
 .0872 8.9403 
 
 .9962 9.9983 
 
 .0875 8.9420 
 
 11.430 
 
 1.0580 
 
 85° 00' 
 
 1.4835 
 
 0.0902 
 
 10 
 
 .0901 .9545 
 
 .9959 .9982 
 
 .0904 .9563 
 
 11.059 
 
 .0«7 
 
 50 
 
 1.4806 
 
 0.0931 
 
 20 
 
 .0929 .9682 
 
 .9957 .9981 
 
 .0934 .9701 
 
 ]0.712 
 
 .0299 
 
 40 
 
 1.4777 
 
 0.0960 
 
 30 
 
 .0958 .9816 
 
 .9954 .9980 
 
 .0963 .9836 
 
 10.385 
 
 .0164 
 
 30 
 
 1.4748 
 
 0.0989 
 
 40 
 
 .0987 .9945 
 
 .9951 .9979 
 
 .0992 .9966 
 
 10.078 
 
 .0034 
 
 20 
 
 1.4719 
 
 0.1018 
 
 50 
 
 .1016 9.0070 
 
 .9948 .9977 
 
 .1022 9.0093 
 
 9.7882 0.9907 
 
 10 
 
 1.4690 
 
 0.1047 
 
 6° 00' 
 
 .1045 9.0192 
 
 .9945 9.9976 
 
 .1051 9.0216 
 
 9.5144 
 
 0.9784 
 
 84° 00' 
 
 1.4661 
 
 0.1076 
 
 10 
 
 1074 .0311 
 
 .9942 .9975 
 
 .1080 .0336 
 
 9.2553 
 
 .9664 
 
 50 
 
 1.4632 
 
 0.1105 
 
 20 
 
 .1103 .0426 
 
 .9939 .9973 
 
 .1110 .0453 
 
 9.0098 
 
 .9547 
 
 40 
 
 1.4603 
 
 0.1134 
 
 30 
 
 .1132 .0539 
 
 .9936 .9972 
 
 .1139 .0567 
 
 8.7769 
 
 .9433 
 
 30 
 
 1.4574 
 
 0.1164 
 
 40 
 
 .1161 .0648 
 
 .9932 .9971 
 
 .1169 .0678 
 
 8.5555 
 
 .9322 
 
 20 
 
 1.4544 
 
 0.1193 
 
 50 
 
 .1190 .0755 
 
 .9929 .9969 
 
 .1198 .0786 
 
 8.3450 
 
 .9214 
 
 10 
 
 1.4515 
 
 0.1222 
 
 7° 00' 
 
 .1219 9.0859 
 
 .9925 9.9968 
 
 .1228 9.0891 
 
 8.1443 
 
 0.9109 
 
 83° 00' 
 
 1.4486 
 
 0.1251 
 
 10 
 
 .1248 .0961 
 
 .9922 .9966 
 
 .1257 .0995 
 
 7.9530 
 
 .9005 
 
 50 
 
 1.4457 
 
 0.1280 
 
 20 
 
 .1276 .1060 
 
 .9918 .9964 
 
 .1287 .1096 
 
 7.7704 
 
 .8904 
 
 40 
 
 1.4428 
 
 0.1309 
 
 30 
 
 .1305 .1157 
 
 .9914 .9963 
 
 .1317 .1194 
 
 7.5958 
 
 .8806 
 
 30 
 
 1.4399 
 
 0.1338 
 
 40 
 
 .1334 .1252 
 
 .9911 .9961 
 
 .1346 .1291 
 
 7.4287 
 
 .8709 
 
 20 
 
 1.4370 
 
 0.1367 
 
 50 
 
 .1363 .1345 
 
 .9907 .9959 
 
 .1376 .1385 
 
 7.2687 
 
 .8615 
 
 10 
 
 1.4341 
 
 0.1396 
 
 8° 00' 
 
 .1392 9.1436 
 
 .9903 9.9958 
 
 .1405 9.1478 
 
 7.1154 
 
 0.8522 
 
 82° 00' 
 
 1.4312 
 
 0.1425 
 
 10 
 
 .1421 .1525 
 
 .9899 .9956 
 
 .1435 .1569 
 
 6.9682 
 
 .8431 
 
 50 
 
 1.4283 
 
 0.1454 
 
 20 
 
 .1449 .1612 
 
 .9894 .9954 
 
 .1465 .1658 
 
 6.8269 
 
 .8342 
 
 40 
 
 1.4254 
 
 0.1484 
 
 30 
 
 .1478 .1697 
 
 .9890 .9952 
 
 .1495 .1745 
 
 6.6912 
 
 .8255 
 
 30 
 
 1.4224 
 
 0.1513 
 
 40 
 
 .1507 .1781 
 
 .9886 .9950 
 
 .1524 .1831 
 
 6.5606 
 
 .8169 
 
 20 
 
 1.4195 
 
 0.1542 
 
 50 
 
 .1536 .1863 
 
 .9881 .9948 
 
 .1554 .1915 
 
 6.4348 
 
 .8085 
 
 10 
 
 1.4166 
 
 0.1571 
 
 9° 00' 
 
 .1564 9.1943 
 
 .9877 9.9946 
 
 .1584 9.1997 
 
 6.3138 0.8003 
 
 81° 00' 
 
 1.4137 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. 
 
 Log. 
 
 
 ■^ 
 
 
 
 COSINES. 
 
 SINES. 
 
 COTANGENTS. 
 
 TANGENTS. 
 
 DEGREES. 
 
 RADIANS. 
 
TABLES. 
 
 147 
 
 Trigonometric Functions. 
 
 RADIANS. 
 
 DEGREES. 
 
 SINES. 
 
 COSINES. 
 
 TANGENTS. 
 
 COTANGENTS. 
 
 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 
 
 0.1571 
 
 9° 00' 
 
 .1564 9.1943 
 
 .9877 9.9946 
 
 .1584 9.1997 
 
 6.3138 0.8003 
 
 81° 00' 
 
 1.4137 
 
 0.1600 
 
 10 
 
 .1593 .2022 
 
 .9872 .9944 
 
 .1614 .2078 
 
 6.1970 .7922 
 
 50 
 
 1.4108 
 
 0.1629 
 
 20 
 
 .1622 .2100 
 
 .9868 .9942 
 
 .1644 .2158 
 
 6.0S44 .7842 
 
 40 
 
 1.4079 
 
 0.1658 
 
 30 
 
 .1650 .2176 
 
 .9863 .9940 
 
 .1673 .2236 
 
 5.9758 .7764 
 
 30 
 
 1.4050 
 
 0.1687 
 
 40 
 
 .1679 .2251 
 
 .9858 .9938 
 
 .1703 .2313 
 
 5.8708 .7687 
 
 20 
 
 1.4021 
 
 0.1716 
 
 50 
 
 .1708 .2324 
 
 .9853 .9936 
 
 .1733 .2389 
 
 5.7694 .7611 
 
 10 
 
 1.3992 
 
 0.1745 
 
 10° 00' 
 
 .1736 9.2397 
 
 .9848 9.9934 
 
 .1763 9.2463 
 
 5.6713 0.7537 
 
 80° 00' 
 
 1.3963 
 
 0.1774 
 
 10 
 
 .1765 .2468 
 
 .9843 .9931 
 
 .1793 .2536 
 
 5.5764 .7464 
 
 50 
 
 1.3934 
 
 0.1804 
 
 20 
 
 .1794 .2538 
 
 .9838 .9929 
 
 .1823 .2609 
 
 5.4845 .7391 
 
 40 
 
 1.3904 
 
 0.1833 
 
 30 
 
 .1822 .2606 
 
 .9833 .9927 
 
 .1853 .2680 
 
 5.3955 .7320 
 
 30 
 
 1.3875 
 
 0.1862 
 
 40 
 
 .1851 .2674 
 
 .9827 .9924 
 
 .1883 .2750 
 
 5.3093 .7250 
 
 20 
 
 1.3846 
 
 0.1891 
 
 50 
 
 .1880 .2740 
 
 .9822 .9922 
 
 .1914 .2819 
 
 5.2257 .7181 
 
 10 
 
 1.3817 
 
 0.1920 
 
 11° 00' 
 
 .1908 9.2806 
 
 .9816 9.9919 
 
 .1944 9.2S87 
 
 5.1446 0.7113 
 
 79° 00' 
 
 1.3788 
 
 0.1949 
 
 10 
 
 .1937 .2870 
 
 .9811 .9917 
 
 .1974 .2953 
 
 5.0658 .7047 
 
 50 
 
 1.3759 
 
 0.1978 
 
 20 
 
 .1965 .2934 
 
 .9805 .9914 
 
 .2004 .3020 
 
 4.9894 .6980 
 
 40 
 
 1.3730 
 
 0.2007 
 
 30 
 
 .1994 .2997 
 
 .9799 .9912 
 
 .2035 .3085 
 
 4.9152 .6915 
 
 30 
 
 1.3701 
 
 0.2036 
 
 40 
 
 .2022 .3058 
 
 .9793 .9909 
 
 .2065 .3149 
 
 4.8430 .6851 
 
 20 
 
 1.3672 
 
 0.2065 
 
 50 
 
 .2051 .3119 
 
 .9787 .9907 
 
 .2095 .3212 
 
 4.7729 .6788 
 
 10 
 
 1.3643 
 
 0.2094 
 
 12° 00' 
 
 .2079 9.3179 
 
 .9781 9.9904 
 
 .2126 9.3275 
 
 4.7046 0.6725 
 
 78° 00' 
 
 1.3614 
 
 0.2123 
 
 10 
 
 .2108 .3238 
 
 .9775 .9901 
 
 .2156 .3336 
 
 4.6382 .6664 
 
 50 
 
 1.3584 
 
 0.2153 
 
 20 
 
 .2136 .3296 
 
 .9769 .9899 
 
 .2186 .3397 
 
 4.5736 .6603 
 
 40 
 
 1.3555 
 
 0.2182 
 
 30 
 
 .2164 .3353 
 
 .9763 .9896 
 
 .2217 .3458 
 
 4.5107 .6542 
 
 30 
 
 1.3526 
 
 0.2211 
 
 40 
 
 .2193 .3410 
 
 .9757 .9893 
 
 .2247 .3517 
 
 4.4494 .6483 
 
 20 
 
 1.3497 
 
 0.2240 
 
 50 
 
 .2221 .3466 
 
 .9750 .9890 
 
 .2278 .3576 
 
 4.3897 .6424 
 
 10 
 
 1.3468 
 
 0.2269 
 
 13° 00' 
 
 .2250 9.3521 
 
 .9744 9.9887 
 
 .2309 9.3634 
 
 4.3315 0.6366 
 
 77° 00' 
 
 1.3439 
 
 0.2298 
 
 10 
 
 .2278 .3575 
 
 .9737 .9884 
 
 .2339 .3691 
 
 4.2747 .6309 
 
 50 
 
 1.3410 
 
 0.2327 
 
 20 
 
 .2306 .3629 
 
 .9730 .9881 
 
 .2370 .3748 
 
 4.2193 .6252 
 
 40 
 
 1.3381 
 
 0.2356 
 
 30 
 
 .2334 .3682 
 
 .9724 .9878 
 
 .2401 .3804 
 
 4.1653 .6196 
 
 30 
 
 1.3352 
 
 0.2385 
 
 40 
 
 .2363 .3734 
 
 .9717 .9875 
 
 .2432 .3859 
 
 4.1126 .6141 
 
 20 
 
 1.3323 
 
 0.2414 
 
 50 
 
 .2391 .3786 
 
 .9710 .9872 
 
 .2462 .3914 
 
 4.0611 .6086 
 
 10 
 
 1.3294 
 
 0.2443 
 
 14° 00' 
 
 .2419 9.3837 
 
 .9703 9.9869 
 
 .2493 9.3968 
 
 4.0108 0.6032 
 
 76° 00' 
 
 1.3265 
 
 0.2473 
 
 10 
 
 .2447 .3887 
 
 .9696 .9866 
 
 .2524 .4021 
 
 3.9617 .5979 
 
 50 
 
 1.3235 
 
 0.2502 
 
 20 
 
 .2476 .3937 
 
 .9689 .9863 
 
 .2555 .4074 
 
 3.9136 .5926 
 
 40 
 
 1.3206 
 
 0.2531 
 
 30 
 
 .2504 .3986 
 
 .9681 .9859 
 
 .2586 .4127 
 
 3.8667 .5873 
 
 30 
 
 1.3177 
 
 0.2560 
 
 40 
 
 .2532 .4035 
 
 .9674 .9856 
 
 .2617 .4178 
 
 3.8208 .5822 
 
 20 
 
 1.3148 
 
 0.2589 
 
 50 
 
 .2560 .4083 
 
 .9667 .9853 
 
 .2648 .4230 
 
 3.7760 .5770 
 
 10 
 
 1.3119 
 
 0.2618 
 
 15° 00' 
 
 .2588 9.4130 
 
 .9659 9.9849 
 
 .2679 9.4281 
 
 3.7321 0.5719 
 
 75° 00' 
 
 1.3090 
 
 0.2647 
 
 10 
 
 .2616 .4177 
 
 .9652 .9846 
 
 .2711 .4331 
 
 3.6891 .5669 
 
 50 
 
 1.3061 
 
 0.2676 
 
 20 
 
 .2644 .4223 
 
 .9644 .9843 
 
 .2742 .4381 
 
 3.6470 .5619 
 
 40 
 
 1.3032 
 
 0.2705 
 
 30 
 
 .2672 .4269 
 
 .9636 .9839 
 
 .2773 .4430 
 
 3.6059 .5570 
 
 30 
 
 1.3003 
 
 0.2734 
 
 40 
 
 .2700 .4314 
 
 .9628 .9836 
 
 .2805 .4479 
 
 3.5656 .5521 
 
 20 
 
 1.2974 
 
 0.2763 
 
 50 
 
 .2728 .4359 
 
 .9621 .9832 
 
 .2836 .4527 
 
 3.5261 .5473 
 
 10 
 
 1.2945 
 
 0.2793 
 
 16° 00' 
 
 .2756 9.4403 
 
 .9613 9.9828 
 
 .2867 9.4575 
 
 3.4874 0.5425 
 
 74° 00' 
 
 1.2915 
 
 0.2822 
 
 10 
 
 .2784 .4447 
 
 .9605 .9825 
 
 .2899 .4622 
 
 3.4495 .5378 
 
 50 
 
 1.2886 
 
 0.2851 
 
 20 
 
 .2812 .4491 
 
 .9596 .9821 
 
 .2931 .4669 
 
 3.4124 .5331 
 
 40 
 
 1.2857 
 
 0.2880 
 
 30 
 
 .2840 .4533 
 
 .9588 .9817 
 
 .2962 .4716 
 
 3.3759 .5284 
 
 30 
 
 1.2828 
 
 0.2909 
 
 40 
 
 .2868 .4576 
 
 .9580 .9814 
 
 .2994 .4762 
 
 3.3402 .5238 
 
 20 
 
 1.2799 
 
 0.2938 
 
 50 
 
 .2896 .4618 
 
 .9572 .9810 
 
 .3026 .4808 
 
 3.3052 .5192 
 
 10 
 
 1.2770 
 
 0.2967 
 
 17° 00' 
 
 .2924 9.4659 
 
 .9563 9.9806 
 
 .3057 9.4853 
 
 3.2709 0.5147 
 
 73° 00' 
 
 1.2741 
 
 0.2996 
 
 10 
 
 .2952 .4700 
 
 .9555 .9802 
 
 .3089 .4898 
 
 3.2371 .5102 
 
 50 
 
 1.2712 
 
 0.3025 
 
 20 
 
 .2979 .4741 
 
 .9546 .9798 
 
 .3121 .4943 
 
 3.2041 .5057 
 
 40 
 
 1.2683 
 
 0.3054 
 
 30 
 
 .3007 .4781 
 
 .9537 .9794 
 
 .3153 .4987 
 
 3.1716 .5013 
 
 30 
 
 1.2654 
 
 0.3083 
 
 40 
 
 .3035 .4821 
 
 .9528 .9790 
 
 .3185 .5031 
 
 3.1397 .4969 
 
 20 
 
 1.2625 
 
 0.3113 
 
 50 
 
 .3062 .4861 
 
 .9520 .9786 
 
 .3217 .5075 
 
 3.1084 .4925 
 
 10 
 
 1.2595 
 
 0.3142 
 
 18° 00' 
 
 .3090 9.4900 
 
 .9511 9.9782 
 
 .3249 9.5118 
 
 3.0777 0.4882 
 
 72° 00' 
 
 1.2566 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 
 
 
 
 COSINES. 
 
 SINES. 
 
 COTANGENTS. 
 
 TANGENTS. 
 
 DEGREES. 
 
 RADIANS. 
 
148 
 
 TABLES. 
 
 Trigonometric Functions. 
 
 RADIANS. 
 
 DEGREES. 
 
 SINES. 
 
 COSINES. 
 
 TANGENTS. 
 
 COTANGENTS. 
 
 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 i Nat. 
 
 Log. 
 
 
 
 0.3142 
 
 18° 00' 
 
 .3090 9.4900 
 
 .9511 9.9782 
 
 .3249 9.5118 
 
 3.0777 0.4882 
 
 72° 00' 
 
 1.2566 
 
 0.3171 
 
 10 
 
 .3118 .4939 
 
 .9502 .9778 
 
 .3281 .5161 
 
 3.0475 
 
 .4839 
 
 50 
 
 1.2537 
 
 0.3200 
 
 20 
 
 .3145 .4977 
 
 .9492 .9774 
 
 .3314 .5203 
 
 3.0178 
 
 .4797 
 
 40 
 
 1.2508 
 
 0.3229 
 
 30 
 
 .3173 .5015 
 
 .9483 .9770 
 
 .3346 .5245 
 
 ' 2.9887 
 
 .4755 
 
 30 
 
 1.2479 
 
 0.3258 
 
 40 
 
 .3201 .5052 
 
 .9474 .9765 
 
 .3378 .5287 
 
 2.9600 
 
 .4713 
 
 20 
 
 1.2450 
 
 0.3287 
 
 50 
 
 .3228 .5090 
 
 .9465 .9761 
 
 .3411 .5329 
 
 2.9319 
 
 .4671 
 
 10 
 
 1.2421 
 
 0.3316 
 
 19° 00' 
 
 .3256 9.5126 
 
 .9455 9.9757 
 
 .3443 9.5370 
 
 2.9042 
 
 0.4630 
 
 71° 00' 
 
 1.2392 
 
 0.3345 
 
 10 
 
 .3283 .5163 
 
 .9446 .9752 
 
 .3476 .5411 
 
 2.8770 
 
 .4589 
 
 50 
 
 1.2363 
 
 0.3374 
 
 20 
 
 .3311 .5199 
 
 .9436 .9748 
 
 .3508 .5451 
 
 2.8502 
 
 .4549 
 
 40 
 
 1.2334 
 
 0.3403 
 
 30 
 
 .3338 .5235 
 
 .9426 .9743 
 
 .3541 .5491 
 
 2.8239 
 
 .4509 
 
 30 
 
 1.2305 
 
 0.3432 
 
 40 
 
 .3365 .5270 
 
 .9417 .9739 
 
 .3574 .5531 
 
 2.7980 
 
 .4469 
 
 20 
 
 1.2275 
 
 0.3462 
 
 50 
 
 .3393 .5306 
 
 .9407 .9734 
 
 .3607 .5571 
 
 2.7725 
 
 .4429 
 
 10 
 
 1.2246 
 
 0.3491 
 
 20° 00' 
 
 .3420 9.5341 
 
 .9397 9.9730 
 
 .3640 9.5611 
 
 2.7475 
 
 0.4389 
 
 70° 00' 
 
 1.2217 
 
 0.3520 
 
 10 
 
 .3448 .5375 
 
 .9387 .9725 
 
 .3673 .5650 
 
 2.7228 
 
 .4350 
 
 SO 
 
 1.2188 
 
 0.3549 
 
 20 
 
 .3475 .5409 
 
 .9377 .9721 
 
 .3706 .5689 
 
 2.6985 
 
 .4311 
 
 40 
 
 1.2159 
 
 0.3578 
 
 30 
 
 .3502 .5443 
 
 .9367 .9716 
 
 .3739 .5727 
 
 2.6746 
 
 .4273 
 
 30 
 
 1.2130 
 
 0.3607 
 
 40 
 
 .3529 .5477 
 
 .9356 .9711 
 
 .3772 .5766 
 
 2.6511 
 
 .4234 
 
 20 
 
 1.2101 
 
 0.3636 
 
 50 
 
 .3557 .5510 
 
 .9346 .9706 
 
 .3805 .5804 
 
 2.6279 
 
 .4196 
 
 10 
 
 1.2072 
 
 0.3665 
 
 21° 00' 
 
 .3584 9.5543 
 
 .9336 9.9702 
 
 .3839 9.5842 
 
 2 6051 
 
 0.4158 
 
 69° 00' 
 
 1.2043 
 
 0.3694 
 
 10 
 
 .3611 .5576 
 
 .9325 .9697 
 
 .3872 .5879 
 
 2.5826 
 
 .4121 
 
 50 
 
 1.2014 
 
 0.3723 
 
 20 
 
 .3638 .5609 
 
 .9315 .9692 
 
 .3906 .5917 
 
 2.5605 
 
 .4083 
 
 40 
 
 1.1985 
 
 0.3752 
 
 30 
 
 .3665 .5641 
 
 .9304 .9687 
 
 .3939 .5954 
 
 25386 
 
 .4046 
 
 30 
 
 1.1956 
 
 0.3782 
 
 40 
 
 .3692 .5673 
 
 .9293 .9682 
 
 .3973 .5991 
 
 2.5172 
 
 .4009 
 
 20 
 
 1.1926 
 
 0.3811 
 
 50 
 
 .3719 .5704 
 
 .9283 .9677 
 
 .4006 .6028 
 
 2.4960 
 
 .3972 
 
 10 
 
 1.1897 
 
 0.3840 
 
 22° 00' 
 
 .3746 9.5736 
 
 .9272 9.9672 
 
 .4040 9.6064 
 
 2.4751 
 
 0.3936 
 
 68° 00' 
 
 1.1868 
 
 0.3869 
 
 10 
 
 .3773 .5767 
 
 .9261 .9667 
 
 .4074 .6100 
 
 2.4545 
 
 .3900 
 
 50 
 
 1.1839 
 
 0.3898 
 
 20 
 
 .3800 .5798 
 
 .9250 .9661 
 
 .4108 .6136 
 
 2.4342 
 
 .3864 
 
 40 
 
 1.1810 
 
 0.3927 
 
 30 
 
 .3827 .5828 
 
 .9239 .9656 
 
 .4142 .6172 
 
 2.4142 
 
 .3828 
 
 30 
 
 1.1781 
 
 0.3956 
 
 40 
 
 .3854 .5859 
 
 .9228 .9651 
 
 .4176 .6208 
 
 2.3945 
 
 .3792 
 
 20 
 
 1.1752 
 
 0.3985 
 
 SO 
 
 .3831 .5889 
 
 .9216 .9646 
 
 .4210 .6243 
 
 23750 
 
 .3757 
 
 10 
 
 1.1723 
 
 0.4014 
 
 23° 00' 
 
 .3907 9.S919 
 
 .9205 9.9640 
 
 .4245 9.6279 
 
 2.3559 0.3721 
 
 67° 00' 
 
 1.1694 
 
 0.4043 
 
 10 
 
 .3934 .5948 
 
 .9194 .9635 
 
 .4279 .6314 
 
 2.3369 
 
 .3686 
 
 SO 
 
 1.1665 
 
 0.4072 
 
 20 
 
 .3961 .5978 
 
 .9182 .9629 
 
 .4314 .6348 
 
 2.3183 
 
 .3652 
 
 40 
 
 1.1636 
 
 0.4102 
 
 30 
 
 .3987 .6007 
 
 .9171 .9624 
 
 .4348 .6383 
 
 2.2998 
 
 .3617 
 
 30 
 
 1.1606 
 
 0.4131 
 
 40 
 
 .4014 , .6036 
 
 .9159 .9618 
 
 .4383 .6417 
 
 2.2817 
 
 .3583 
 
 20 
 
 1.1577 
 
 0.4160 
 
 50 
 
 .4041 .6065 
 
 .9147 .9613 
 
 .4417 .6452 
 
 2.2637 
 
 .3548 
 
 10 
 
 1.1548 
 
 0.4189 
 
 24° 00' 
 
 .4067 9.6093 
 
 .9135 9.9607 
 
 .4452 9.6486 
 
 2.2460 0.3514 
 
 66° 00' 
 
 1.1519 
 
 0.4218 
 
 10 
 
 .4094 .6121 
 
 .9124 .9602 
 
 .4487 .6520 
 
 2.2286 
 
 .3480 
 
 SO 
 
 1.1490 
 
 0.4247 
 
 20 
 
 .4120 .6149 
 
 .9112 .9596 
 
 .4522 .6553 
 
 2.2113 
 
 .3447 
 
 40 
 
 1.1461 
 
 0.4276 
 
 30 
 
 .4147 .6177 
 
 .9100 .9590 
 
 .4557 .6587 
 
 2.1943 
 
 .3413 
 
 30 
 
 1.1432 
 
 0.4305 
 
 40 
 
 .4173 .6205 
 
 .9088 .9584 
 
 .4592 .6620 
 
 2.1775 
 
 .3380 
 
 20 
 
 1.1403 
 
 0.4334 
 
 50 
 
 .4200 .6232 
 
 .9075 .9579 
 
 .4628 .6654 
 
 2.1609 
 
 .3346 
 
 10 
 
 1.1374 
 
 0.4363 
 
 25° 00' 
 
 .4226 9.6259 
 
 .9063 9.9573 
 
 .4663 9.6687 
 
 2 1445 
 
 0.3313 
 
 65° 00' 
 
 1.1345 
 
 0.4392 
 
 10 
 
 .4253 .6286 
 
 .9051 .9567 
 
 .4699 .6720 
 
 2.1283 
 
 .3280 
 
 50 
 
 1.1316 
 
 0.4422 
 
 20 
 
 .4279 .6313 
 
 .9038 .9561 
 
 .4734 .6752 
 
 2.1123 
 
 .3248 
 
 40 
 
 1.1286 
 
 0.4451 
 
 30 
 
 .4305 .6340 
 
 .9026 .9555 
 
 .4770 .6785 
 
 2.0965 
 
 .3215 
 
 30 
 
 1.1257 
 
 0.4480 
 
 40 
 
 .4331 .6366 
 
 .9013 .9549 
 
 .4806 .6817 
 
 2.0809 
 
 .3183 
 
 20 
 
 1.1228 
 
 0.4509 
 
 50 
 
 .4358 .6392 
 
 .9001 .9543 
 
 .4841 .6850 
 
 2.0655 
 
 .3150 
 
 10 
 
 1.1199 
 
 0.4538 
 
 26° 00' 
 
 .4384 9.6418 
 
 .8988 9.9537 
 
 .4877 9.6882 
 
 2.0503 0.3118 
 
 64° 00' 
 
 1.1170 
 
 0.4567 
 
 10 
 
 .4410 .6444 
 
 .8975 .9530 
 
 .4913 .6914 
 
 2.0353 
 
 .3086 
 
 50 
 
 1.1141 
 
 0.4596 
 
 20 
 
 .4436 .6470 
 
 .8962 .9524 
 
 .4950 .6946 
 
 2.0204 
 
 .3054 
 
 40 
 
 1.1112 
 
 0.4625 
 
 30 
 
 .4462 .6495 
 
 .8949 .9518 
 
 .4986 .6977 
 
 2.0057 
 
 .3023 
 
 30 
 
 1.1083 
 
 0.4654 
 
 40 
 
 .4488 .6521 
 
 .8936 .9512 
 
 .5022 .7009 
 
 1.9912 
 
 .2991 
 
 20 
 
 1.1054 
 
 0.4683 
 
 SO 
 
 .4514 .6546 
 
 .8923 .9505 
 
 .5059 .7040 
 
 1.9768 
 
 .2960 
 
 10 
 
 1.1025 
 
 0.4712 
 
 27° 00' 
 
 .4540 9.6570 
 
 .8910 9.9499 
 
 .5095 9.7072 
 
 1.9626 0.2928 
 
 63° 00' 
 
 1.0996 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. 
 
 Log. 
 
 
 
 
 
 COSINES. 
 
 SINES. 
 
 COTANGENTS. 
 
 TANGENTS. 
 
 DEGREES. 
 
 RADIANS. 
 
TABLES. 
 
 149 
 
 
 
 
 Trigonometric Functions. 
 
 
 
 
 RADIANS. 
 
 DEGREES. 
 
 SINES. 
 
 COSINES. 
 
 TANGENTS. 
 
 COTANGENTS. 
 
 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. 
 
 Log. 
 
 Nat. Log. 
 
 
 
 0.4712 
 
 27° 00' 
 
 .4540 9.6570 
 
 .8910 9.9499 
 
 .5095 
 
 9.7072 
 
 1.9626 0.2928 
 
 63° 00' 
 
 1.0996 
 
 0.4741 
 
 10 
 
 .4566 .6595 
 
 .8897 .9492 
 
 .5132 
 
 .7103 
 
 1.9486 .2897 
 
 50 
 
 1.0966 
 
 0.4771 
 
 20 
 
 .4592 .6620 
 
 .8884 .9486 
 
 .5169 
 
 .7134 
 
 1.9347 .2866 
 
 40 
 
 1.0937 
 
 0.4800 
 
 30 
 
 .4617 .6644 
 
 .8870 .9479 
 
 .5206 
 
 ,7165 
 
 1.9210 .2835 
 
 30 
 
 1.0908 
 
 0.4829 
 
 40 
 
 .4643 .6668 
 
 .8857 .9473 
 
 .5243 
 
 .7196 
 
 1.9074 .2804 
 
 20 
 
 1.0879 
 
 0.4858 
 
 50 
 
 .4669 .6692 
 
 .8843 .9466 
 
 .5280 
 
 .7226 
 
 1.8940 .2774 
 
 10 
 
 1.0850 
 
 0.4887 
 
 28° 00' 
 
 .4695 9.6716 
 
 .8829 9.9459 
 
 .5317 9.7257 
 
 1.8807 0.2743 
 
 62° 00' 
 
 1.0821 
 
 0.4916 
 
 10 
 
 .4720 .6740 
 
 .8816 .9453 
 
 .5354 
 
 .7287 
 
 1.8676 .2713 
 
 50 
 
 1.0792 
 
 0.4945 
 
 20 
 
 .4746 .6763 
 
 .8802 .9446 
 
 .5392 
 
 .7317 
 
 1.8546 .2683 
 
 40 
 
 1.0763 
 
 0.4974 
 
 30 
 
 .4772 .6787 
 
 .8788 .9439 
 
 .5430 
 
 .7348 
 
 1.8418 .2652 
 
 30 
 
 1.0734 
 
 0.5003 
 
 40 
 
 .4797 .6810 
 
 .8774 .9432 
 
 .5467 
 
 .7378 
 
 1.8291 .2622 
 
 20 
 
 1.0705 
 
 0.5032 
 
 50 
 
 .4823 .6833 
 
 .8760 .9425 
 
 .5505 
 
 .7408 
 
 1.S165 .2592 
 
 10 
 
 1.0676 
 
 0.5061 
 
 29° 00' 
 
 .4848 9.6856 
 
 .8746 9.9418 
 
 .5543 9.7438 
 
 1.8040 0.2562 
 
 61° 00' 
 
 1.0647 
 
 0.5091 
 
 10 
 
 .4874 .6878 
 
 .8732 .9411 
 
 .5581 
 
 .7467 
 
 1.7917 .2533 
 
 50 
 
 1.0617 
 
 0.5120 
 
 20 
 
 .4899 .6901 
 
 .8718 .9404 
 
 .5619 
 
 .7497 
 
 1.7796 .2503 
 
 40 
 
 1.0588 
 
 0.5149 
 
 30 
 
 .4924 .6923 
 
 .8704 .9397 
 
 .5658 
 
 .7526 
 
 1.7675 .2474 
 
 30 
 
 1.0559 
 
 0.5178 
 
 40 
 
 .4950 .6946 
 
 .8689 .9390 
 
 .5696 
 
 .7556 
 
 1.7556 .2444 
 
 20 
 
 1.0530 
 
 0.5207 
 
 50 
 
 .4975 .6968 
 
 .8675 .9383 
 
 .5735 
 
 .7585 
 
 1.7437 .2415 
 
 10 
 
 1.0501 
 
 0.5236 
 
 30° 00' 
 
 .5000 9.6990 
 
 .8660 9.9375 
 
 .5774 9.7614 
 
 1.7321 0.2386 
 
 60° 00' 
 
 1.0472 
 
 0.5265 
 
 10 
 
 .5025 .7012 
 
 .8646 .9368 
 
 .5812 
 
 .7644 
 
 1.7205 .2356 
 
 50 
 
 1.0443 
 
 0.5294 
 
 20 
 
 .5050 .7033 
 
 .8631 .9361 
 
 .5851 
 
 .7673 
 
 1.7090 .2327 
 
 40 
 
 1.0414 
 
 0.5323 
 
 30 
 
 .5075 .7055 
 
 .8616 .9353 
 
 .5890 
 
 .7701 
 
 1.6977 .2299 
 
 30 
 
 1.0385 
 
 0.5352 
 
 40 
 
 .5100 .7076 
 
 .8601 .9346 
 
 .5930 
 
 .7730 
 
 1.6864 .2270 
 
 20 
 
 1.0356 
 
 0.5381 
 
 50 
 
 .5125 .7097 
 
 .8587 .9338 
 
 .5969 
 
 .7759 
 
 1.6753 .2241 
 
 10 
 
 1.0327 
 
 0.5411 
 
 31° 00' 
 
 .5150 9.7118 
 
 .8572 9.9331 
 
 .6009 9.7788 
 
 1.6643 0.2212 
 
 59° 00' 
 
 1.0297 
 
 0.5440 
 
 10 
 
 .5175 .7139 
 
 .8557 .9323 
 
 .6048 
 
 .7816 
 
 1.6534 .2184 
 
 50 
 
 1.0268 
 
 0.5469 
 
 20 
 
 .5200 .7160 
 
 .8542 .9315 
 
 .6088 
 
 .7845 
 
 1.6426 .2155 
 
 40 
 
 1.0239 
 
 0.5498 
 
 30 
 
 .5225 .7181 
 
 .8526 .9.308 
 
 .6128 
 
 .7873 
 
 1.6319 .2127 
 
 30 
 
 1.0210 
 
 0.5527 
 
 40 
 
 .5250 .7201 
 
 .8511 .9300 
 
 .6168 
 
 .7902 
 
 1.6212 .2098 
 
 20 
 
 1.0181 
 
 0.5556 
 
 50 
 
 .5275 .7222 
 
 .8496 .9292 
 
 .6208 
 
 .7930 
 
 1.6107 .2070 
 
 10 
 
 1.0152 
 
 0.5585 
 
 32° 00' 
 
 .5299 9.7242 
 
 .8480 9.9284 
 
 .6249 9.7958 
 
 1.6003 0.2042 
 
 58° 00' 
 
 1.0123 
 
 0.5614 
 
 10 
 
 .5324 .7262 
 
 .8465 .9276 
 
 .6289 
 
 .7986 
 
 1.5900 .2014 
 
 50 
 
 1.0094 
 
 0.5643 
 
 20 
 
 .5348 .7282 
 
 .8450 .9268 
 
 .6330 
 
 .8014 
 
 1.5798 .1986 
 
 40 
 
 1.0065 
 
 0.5672 
 
 30 
 
 .5373 .7302 
 
 .8434 .9260 
 
 .6371 
 
 .8042 
 
 1.5697 .1958 
 
 30 
 
 1.0036 
 
 0.5701 
 
 40 
 
 .5398 .7322 
 
 .8418 .9252 
 
 .6412 
 
 .8070 
 
 1.5597 .1930 
 
 20 
 
 1.0007 
 
 0.5730 
 
 50 
 
 .5422 .7342 
 
 .8403 .9244 
 
 .6453 
 
 .8097 
 
 1.5497 .1903 
 
 10 
 
 0.9977 
 
 0.5760 
 
 33° 00' 
 
 .5446 9.7361 
 
 .8387 9.9236 
 
 .6494 9.8125 
 
 1.5399 0.1875 
 
 57° 00' 
 
 0.9948 
 
 0.5789 
 
 10 
 
 .5471 .7380 
 
 .8371 .9228 
 
 .6536 
 
 .8153 
 
 1.5301 .1847 
 
 50 
 
 0.9919 
 
 0.5818 
 
 20 
 
 .5495 .7400 
 
 .8355 .9219 
 
 .6577 
 
 .8180 
 
 1.5204 .1820 
 
 40 
 
 0.9890 
 
 0.5847 
 
 30 
 
 .5519 .7419 
 
 .8339 .9211 
 
 .6619 
 
 .8208 
 
 1.5108 .1792 
 
 30 
 
 0.9861 
 
 0.5876 
 
 40 
 
 .5544 .7438 
 
 .8323 .9203 
 
 .6661 
 
 .8235 
 
 1.5013 .1765 
 
 20 
 
 0.9832 
 
 0.5905 
 
 50 
 
 .5568 .7457 
 
 .8307 .9194 
 
 .6703 
 
 .8263 
 
 1.4919 .1737 
 
 10 
 
 0.9803 
 
 0.5934 
 
 34° 00' 
 
 .5592 9.7476 
 
 .8290 9.9186 
 
 .6745 
 
 9.8290 
 
 1.4826 0.1710 
 
 56° 00' 
 
 0.9774 
 
 0.5963 
 
 10 
 
 .5616 .7494 
 
 .8274 .9177 
 
 .6787 
 
 .8317 
 
 1.4733 .1683 
 
 50 
 
 0.9745 
 
 0.5992 
 
 20 
 
 .5640 .7513 
 
 .8258 .9169 
 
 .6830 
 
 .8344 
 
 1.4641 .1656 
 
 40 
 
 0.9716 
 
 0.6021 
 
 30 
 
 .5664 .7531 
 
 .8241 .9160 
 
 .6873 
 
 .8371 
 
 1.4550 .1629 
 
 30 
 
 0.9687 
 
 0.6050 
 
 40 
 
 .5688 .7550 
 
 .8225 .9151 
 
 .6916 
 
 .8398 
 
 1.4460 .1602 
 
 20 
 
 0.9657 
 
 0.6080 
 
 50 
 
 .5712 .7568 
 
 .8208 .9142 
 
 .6959 
 
 .8425 
 
 1.4370 .1575 
 
 10 
 
 0.9628 
 
 0.6109 
 
 35° 00' 
 
 .5736 9.7586 
 
 .8192 9.9134 
 
 .7002 
 
 9.8452 
 
 1.4281 0.1548 
 
 55° 00' 
 
 0.9599 
 
 0.6138 
 
 10 
 
 .5760 .7604 
 
 .8175 .9125 
 
 .7046 
 
 .8479 
 
 1.4193 .1521 
 
 50 
 
 0.9570 
 
 0.6167 
 
 20 
 
 .5783 .7622 
 
 .8158 .9116 
 
 .7089 
 
 .8506 
 
 1.4106 .1494 
 
 40 
 
 0.9541 
 
 0.6196 
 
 30 
 
 .5807 .7640 
 
 .8141 .9107 
 
 .7133 
 
 .8533 
 
 1.4019 .1467 
 
 30 
 
 0.9512 
 
 0.6225 
 
 40 
 
 .5831 .7657 
 
 .8124 .9098 
 
 .7177 
 
 .8559 
 
 1.3934 .1441 
 
 20 
 
 0.9483 
 
 0.6254 
 
 50 
 
 .5854 7675 
 
 .8107 .9089 
 
 .7221 
 
 .8586 
 
 1.3848 .1414 
 
 10 
 
 0.9454 
 
 0.6283 
 
 36° 00' 
 
 .5878 9.7692 
 
 .8090 9.9080 
 
 .7265 
 
 9.8613 
 
 1.3764 0.1387 
 
 54° 00' 
 
 0.9425 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. 
 
 Log. 
 
 Nat. Log. 
 
 
 
 
 
 COSINES. 
 
 SINES. 
 
 COTANGENTS. 
 
 TANGENTS. 
 
 DEGREES. 
 
 RADIANS. 
 
150 
 
 
 
 TABLES. 
 
 
 
 
 
 
 
 Trigonometric Functions. 
 
 
 RADIANS. 
 
 DEGREES. 
 
 SINES. 
 
 COSINES. 
 
 TANGENTS. 
 
 COTANGENTS. 
 
 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 
 
 0.62S3 
 
 36° 00' 
 
 .5878 9.7692 
 
 .8090 9.9080 
 
 .7265 9.8613 
 
 1.3764 0.13S7 
 
 54° 00' 
 
 0.9425 
 
 0.6312 
 
 10 
 
 .5901 .7710 
 
 .8073 .9070 
 
 .7310 .8639 
 
 1.3680 .1361 
 
 50 
 
 0.9396 
 
 0.6341 
 
 20 
 
 .5925 .7727 
 
 .8056 .9061 
 
 .7355 .8666 
 
 1.3597 .1334 
 
 40 
 
 0.9367 
 
 0.6370 
 
 30 
 
 .5948 .7744 
 
 .8039 .9052 
 
 .7400 .8692 
 
 1.3514 .1308 
 
 30 
 
 0.9338 
 
 0.6400 
 
 40 
 
 .5972 .7761 
 
 .8021 .9042 
 
 .7445 .8718 
 
 1.3432 .1282 
 
 20 
 
 0.9308 
 
 0.6429 
 
 50 
 
 .5995 .7778 
 
 .8004 .9033 
 
 .7490 .8745 
 
 1.3351 .1255 
 
 10 
 
 0.9279 
 
 0.6458 
 
 37° 00' 
 
 .6018 9.7795 
 
 .7986 9.9023 
 
 .7536 9.8771 
 
 1.3270 0.1229 
 
 53° 00' 
 
 0.9250 
 
 0.64S7 
 
 10 
 
 .6041 .7811 
 
 .7969 .9014 
 
 .7581 .8797 
 
 1.3190 .1203 
 
 50 
 
 0.9221 
 
 0.6516 
 
 20 
 
 .6065 .7828 
 
 .7951 .9004 
 
 .7627 .8824 
 
 1.3111 .1176 
 
 40 
 
 0.9192 
 
 0.6545 
 
 30 
 
 .6088 .7844 
 
 .7934 .8995 
 
 .7673 .8850 
 
 1.3032 .1150 
 
 30 
 
 0.9163 
 
 0.6574 
 
 40 
 
 .6111 .7861 
 
 .7916 .8985 
 
 .7720 .8876 
 
 1.2954 .1124 
 
 20 
 
 0.9134 
 
 0.6603 
 
 50 
 
 .6134 .7877 
 
 .7898 .8975 
 
 .7766 .8902 
 
 1.2876 .1098 
 
 10 
 
 0.9105 
 
 0.6632 
 
 38° 00' 
 
 .6157 9.7893 
 
 .7880 9.8965 
 
 .7813 9.8928 
 
 1.2799 0.1072 
 
 52° 00' 
 
 0.9076 
 
 0.6661 
 
 10 
 
 .6180 .7910 
 
 .7862 .8955 
 
 .7860 .8954 
 
 1.2723 .1046 
 
 50 
 
 0.9047 
 
 0.6690 
 
 20 
 
 .6202 .7926 
 
 .7844 .8945 
 
 .7907 .8980 
 
 1.2647 .1020 
 
 40 
 
 0.9018 
 
 0.6720 
 
 30 
 
 .6225 .7941 
 
 .7826 .8935 
 
 .7954 .9006 
 
 1.2572 .0994 
 
 30 
 
 0.8988 
 
 0.6749 
 
 40 
 
 .6248 .7957 
 
 .7808 .8925 
 
 .8002 .9032 
 
 1.2497 .0968 
 
 20 
 
 0.8959 
 
 0.67 7S 
 
 50 
 
 .6271 .7973 
 
 .7790 .8915 
 
 .8050 .9058 
 
 1.2423 .0942 
 
 10 
 
 0.8930 
 
 0.6807 
 
 39° 00' 
 
 .6293 9.7989 
 
 .7771 9.8905 
 
 .8098 9.9084 
 
 1.2349 0.0916 
 
 51° 00' 
 
 0.8901 
 
 0.6836 
 
 10 
 
 .6316 .8004 
 
 .7753 .8895 
 
 .8146 .9110 
 
 1.2276 .0890 
 
 50 
 
 0.8872 
 
 0.6865 
 
 20 
 
 .6338 .8020 
 
 .7735 .8884 
 
 .8195 .9135 
 
 1.2203 .0865 
 
 40 
 
 0.8843 
 
 0.6894 
 
 30 
 
 .6361 .8035 
 
 .7716 .8874 
 
 .8243 .9161 
 
 1.2131 .0839 
 
 30 
 
 0.8814 
 
 0.6923 
 
 40 
 
 .6383 .8050 
 
 .7698 .8864 
 
 .8292 .9187 
 
 1.2059 .0813 
 
 20 
 
 0.8785 
 
 0.6952 
 
 50 
 
 .6406 .8066 
 
 .7679 .8853 
 
 .8342 .9212 
 
 1.1988 .0788 
 
 10 
 
 0.8756 
 
 0.6981 
 
 40° 00' 
 
 .6428 9.S081 
 
 .7660 9.8843 
 
 .8391 9.9238 
 
 1.1918 0.0762 
 
 50° 00' 
 
 0.8727 
 
 0.7010 
 
 10 
 
 .6450 .8096 
 
 .7642 .8832 
 
 .8441 .9264 
 
 1.1847 .0736 
 
 50 
 
 0.S698 
 
 0.7039 
 
 20 
 
 .6472 .8111 
 
 .7623 .8821 
 
 .8491 .9289 
 
 1.1778 .0711 
 
 40 
 
 0.8668 
 
 0.7069 
 
 30 
 
 .6494 .8125 
 
 .7604 .8810 
 
 .8541 .9315 
 
 1.1708 .0685 
 
 30 
 
 0.8639 
 
 0.7098 
 
 40 
 
 .65.'.7 .8140 
 
 .7585 .8800 
 
 .8591 .9341 
 
 1.1640 .0659 
 
 20 
 
 0.S610 
 
 0.7127 
 
 50 
 
 .6539 .8155 
 
 .7566 .8789 
 
 .8642 .9366 
 
 1.1571 .0634 
 
 10 
 
 0.8581 
 
 0.7156 
 
 41° 00' 
 
 .6561 9.8169 
 
 .7547 9.8778 
 
 .8693 9.9392 
 
 1.1504 0.0608 
 
 49° 00' 
 
 0.8552 
 
 0.7185 
 
 10 
 
 .6583 .8184 
 
 .7528 .8767 
 
 .8744 .9417 
 
 1.1436 .0583 
 
 50 
 
 0.8523 
 
 0.7214 
 
 20 
 
 .6604 .8198 
 
 .7509 .8756 
 
 .8796 .9443 
 
 1.1369 .0557 
 
 40 
 
 0.8494 
 
 0.7243 
 
 30 
 
 .6626 .8213 
 
 .7490 .8745 
 
 .8847 .9468 
 
 1.1303 .0532 
 
 30 
 
 0.8465 
 
 0.7272 
 
 40 
 
 .6648 .8227 
 
 .7470 .8733 
 
 .8899 .9494 
 
 1.1237 .0506 
 
 20 
 
 0.8436 
 
 0.7301 
 
 50 
 
 .6670 .8241 
 
 .7451 .8722 
 
 .8952 .9519 
 
 1.1171 .0481 
 
 10 
 
 0.8407 
 
 0.7330 
 
 42° 00' 
 
 .6691 9.8255 
 
 .7431 9.8711 
 
 .9004 9.9544 
 
 1.1106 0.0456 
 
 48° 00' 
 
 0.8378 
 
 0.7359 
 
 10 
 
 .6713 .8269 
 
 .7412 .8699 
 
 .9057 .9570 
 
 1.1041 .0430 
 
 50 
 
 0.8348 
 
 0.7389 
 
 20 
 
 .6734 .8283 
 
 .7392 .8688 
 
 .9110 .9595 
 
 1.0977 .0405 
 
 40 
 
 0.8319 
 
 0.7418 
 
 30 
 
 .6756 .8297 
 
 .7373 .8676 
 
 .9163 .9621 
 
 1.0913 .0379 
 
 30 
 
 0.8290 
 
 0.7447 
 
 40 
 
 .6777 .8311 
 
 .7353 .8665 
 
 .9217 .9646 
 
 1.0850 .0354 
 
 20 
 
 0.8261 
 
 0.7476 
 
 50 
 
 .6799 .8324 
 
 .7333 .8653 
 
 .9271 .9671 
 
 1.0786 .0329 
 
 10 
 
 0.8232 
 
 0.7505 
 
 43° 00' 
 
 .6820 9.8338 
 
 .7314 9.8641 
 
 .9325 9.9697 
 
 1.0724 0.0303 
 
 47° 00' 
 
 0.8203 
 
 0.7534 
 
 10 
 
 .6841 .8351 
 
 .7294 .8629 
 
 .9380 .9722 
 
 1.0661 .0278 
 
 50 
 
 0.8174 
 
 0.7563 
 
 20 
 
 .6862 .8365 
 
 .7274 .8618 
 
 .9435 .9747 
 
 1.0599 .0253 
 
 40 
 
 0.8145 
 
 0.7592 
 
 30 
 
 .6884 .8378 
 
 .7254 .8606 
 
 .9490 .9772 
 
 1.0538 .0228 
 
 30 
 
 0.8116 
 
 0.7621 
 
 40 
 
 .6905 .8391 
 
 .7234 .8594 
 
 .9545 .9798 
 
 1.0477 .0202 
 
 20 
 
 0.8087 
 
 0.7650 
 
 50 
 
 .6926 .8405 
 
 .7214 .8582 
 
 .9601 .9823 
 
 l.ail6 .0177 
 
 10 
 
 0.8058 
 
 0.7679 
 
 44° 00' 
 
 .6947 9.8418 
 
 .7193 9.8569 
 
 .9657 9.9848 
 
 1.0355 0.0152 
 
 46° 00' 
 
 0.8029 
 
 0.7709 
 
 10 
 
 .6967 .8431 
 
 .7173 .8557 
 
 .9713 .9874 
 
 1.0295 .0126 
 
 50 
 
 0.7999 
 
 0.7738 
 
 20 
 
 .6988 .8444 
 
 .7153 .8545 
 
 .9770 .9899 
 
 1.0235 .0101 
 
 40 
 
 0.7970 
 
 0.7767 
 
 30 
 
 .7009 .8457 
 
 .7133 .8532 
 
 .9827 .9924 
 
 1.0176 .0076 
 
 30 
 
 0.7941 
 
 0.7796 
 
 40 
 
 .7030 .8469 
 
 .7112 .8520 
 
 .9884 .9949 
 
 1.0117 .0051 
 
 20 
 
 0.7912 
 
 0.7825 
 
 50 
 
 .7050 .8482 
 
 .7092 .8507 
 
 .9942 .9975 
 
 1.0058 .0025 
 
 10 
 
 0.7883 
 
 0.7854 
 
 45° 00' 
 
 .7071 9.8495 
 
 .7071 9.8495 
 
 1.0000 0.0000 
 
 1.0000 0.0000 
 
 45° 00' 
 
 0.7854 
 
 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 Nat. Log. 
 
 
 
 
 
 COSINES. 
 
 SINES. 
 
 COTANGENT.S. 
 
 TANGENTS. 
 
 DEGREES. 
 
 RADIANS. 
 
TABLES. 
 
 151 
 
 Equivalents of Radians in Degrees, Minutes, and Seconds of Arc. 
 
 RAmATJS- 
 
 EQUIVALENTS. 
 
 RADIANS. 
 
 EQUIVALENTS. 
 
 0.0001 
 
 0° 0' 20".6 or 0°.005730 
 
 0.0600 
 
 3° 26' 15".9 
 
 or 3°.437747 
 
 0.0002 
 
 0° 0'41".3 or 0°.011459 
 
 0.0700 
 
 4° 0'3S".5 
 
 or 4°.010705 
 
 0.0003 
 
 0° 1'01".9 or 0°.017189 
 
 O.OSOO 
 
 4°35'01".2 
 
 or 4°.5S3662 
 
 0.0004 
 
 0° V 22".5 or 0°. 022918 
 
 0.0900 
 
 5° 9'23".S 
 
 or 5°.156620 
 
 0.0005 
 
 0° 1' 43". 1 or 0°.028648 
 
 0.1000 
 
 5° 43' 46". 5 
 
 or 5°.729578 
 
 0.0006 
 
 0° 2'03".8 or 0°.034377 
 
 0.2000 
 
 11°27'33".0 
 
 or 11°.459156 
 
 0.0007 
 
 Qo 2'24".4 or 0°.040107 
 
 0.3000 
 
 17°11'19".4 
 
 or 17°. 188734 
 
 0.0008 
 
 0° 2'45".0 or 0°.045837 
 
 0.4000 
 
 22°55'05".9 
 
 or 22°.918312 
 
 0.0009 
 
 0° 3'05".6 or 0°.051566 
 
 0.5000 
 
 2S°38'52".4 
 
 or 28°.647890 
 
 0.0010 
 
 0° 3'26".3 or 0°. 057296 
 
 0.6000 
 
 34° 22' 3S".9 
 
 or 34°377468 
 
 0.0020 
 
 0° 6'52".5 or 0°.114S92 
 
 0.7000 
 
 40° 6'25".4 
 
 or 40°. 107046 
 
 0.0030 
 
 0°10'1S".8 or 0°.171887 
 
 O.SOOO 
 
 45°50'11".8 
 
 or 45°.836624 
 
 0.0040 
 
 0°13'45".l or 0°. 229183 
 
 0.9000 
 
 51° 33' 58".3 
 
 or 51°.566202 
 
 0.0050 
 
 0°17'11".3 or 0°.286479 
 
 1.0000 
 
 57°17'44".8 
 
 or 57°.295780 
 
 0.0060 
 
 0°20'37".6 or 0^.343775 
 
 2.0000 
 
 114° 35' 29".6 
 
 or 1M°.591559 
 
 0.0070 
 
 0°24'03".9 or 0°.401070 
 
 3.0000 
 
 171° 53' 14".4 
 
 or 171°.8S7339 
 
 0.00S0 
 
 0°27'30".l or 0°. 458366 
 
 4.0000 
 
 229° 10' 59".2 
 
 or 229°. 183 118 
 
 0.0090 
 
 0°30'S6".4 or 0°.515662 
 
 5.0000 
 
 286°28'44".0 
 
 or 286°.478898 
 
 0.0100 
 
 0°34'22".6 or 0°.572958 
 
 6.0000 
 
 343°46'28".8 
 
 or 343°.774677 
 
 0.0200 
 
 1° 8'45".3 or P.145916 
 
 7.0000 
 
 401° 4' 13" 6 
 
 or 401°.070457 
 
 0.0300 
 
 1''43'07".9 or 10.718873 
 
 8.0000 
 
 458° 21' 58".4 
 
 or 458°.366236 
 
 0.0400 
 
 2°17'30".6 or 2°.291831 
 
 9.0000 
 
 515°39'43".3 
 
 or 515°.662016 
 
 0.0500 
 
 2° 51' S3".2 or 2°.864789 
 
 10.0000 
 
 572° 57' 2S".l 
 
 or 572°.95779S 
 
 The Values in Circular Measure of Angles which are given in 
 Degrees and Minutes. 
 
 r 
 
 0.0003 
 
 9' 
 
 0.0026 
 
 3° 
 
 0.0524 
 
 20° 
 
 0.3491 
 
 100° 
 
 \.7453 
 
 2' 
 
 0.0006 
 
 10' 
 
 0.0029 
 
 4° 
 
 0.0698 
 
 30° 
 
 0.5236 
 
 110° 
 
 1.9199 
 
 3' 
 
 0.0009 
 
 20' 
 
 0.0058 
 
 5° 
 
 0.0873 
 
 40° 
 
 0.6981 
 
 120° 
 
 2.0944 
 
 4' 
 
 0.0012 
 
 30' 
 
 0.0087 
 
 6° 
 
 0.1047 
 
 50° 
 
 0.8727 
 
 130° 
 
 2.2689 
 
 5' 
 
 0.0015 
 
 40' 
 
 0.0116 
 
 7° 
 
 0.1222 
 
 60° 
 
 1.0472 
 
 140° 
 
 2.4435 
 
 6' 
 
 0.0017 
 
 50' 
 
 0.0145 
 
 8° 
 
 0.1396 
 
 70° 
 
 1.2217 
 
 150° 
 
 2.6180 
 
 7' 
 
 0.0020 
 
 1° 
 
 0.0175 
 
 9° 
 
 0.1571 
 
 80° 
 
 1.3963 
 
 160° 
 
 2.7925 
 
 8' 
 
 0.0023 
 
 2° 
 
 0.0349 
 
 10° 
 
 0.1745 
 
 90° 
 
 1.5708 
 
 170° 
 
 2.9671 
 
PAGE INDEX. 
 
 INTEGRALS. 
 
 Fundamental forms 
 
 Rational algebraic expressions involving (a + bx) and (a' + b'x) 
 
 {a + bx") . 
 " " " (a + bx + cx^) . 
 
 (a' + 6'x)and(a + 6a; + cx2) 
 Rational fractions ......... 
 
 Irrational algebraic expressions involving Va + bx or \/a + bx . 
 
 (1 
 
 
 11 
 
 Miscellaneous algebraic expressions 
 
 General transcendental forms ...... 
 
 Expressions involving simple direct trigonometric functions 
 Expressions involving inverse trigonometric functions 
 Exponential forms ..... 
 
 Logarithmic forms ..... 
 
 Expressions involving hyperbolic functions 
 
 Miscellaneous definite integrals 
 
 Elliptic integrals 
 
 Pages 
 
 3,4 
 
 5,7 
 
 8,9 
 
 10,11 
 
 11-13 
 
 13,14 
 
 16,17 
 
 18,19 
 
 20-23 
 
 31 
 
 23-27 
 
 (a' + b'x) and Va + bx + cx^ 27-30 
 
 32-34 
 35-37 
 38-51 
 51-53 
 63-56 
 66-58 
 68-61 
 62-65 
 66-72 
 
 V a + bx and Va' + b' x 
 •V x2 ± a2 or Va2 — x^ 
 
 V2 
 
 ax — x^ 
 
 Va + bx + cx2 
 
 AUXILIARY FORMULAS AND TABLES. 
 
 Trigonometric functions . 
 
 Hyperbolic functions 
 
 Elliptic functions, Bessel's functions 
 
 Series 
 
 Derivatives .... 
 
 Green's Theorem and allied formulas 
 
 Table of mathematical constants 
 
 General formulas of integration 
 
 Note on interpolation 
 
 Table of the probability integral 
 
 Tables of elliptic integrals 
 
 Table of hyperbolic functions . 
 
 Table of values of e-^ 
 
 Table of common logarithms of e^ and e-^ 
 
 Five-place table of natural logarithms 
 
 Table of logarithms of T {x) . 
 
 Three-place table of natural trigonometric functions 
 
 Four-place table of common logarithms of numbers 
 
 Four-place table of trigonometric functions 
 
 Tables for reducing radians to degrees 
 
 152 
 
 73-80 
 
 81-83 
 
 84-87 
 
 88-96 
 
 97-106 
 
 106-109 
 
 109 
 
 110-114 
 
 115 
 
 116-120 
 
 121-123 
 
 124-127 
 
 127 
 
 128, 129 
 
 130-139 
 
 140 
 
 141 
 
 142-145 
 
 146-150 
 
 151 
 
14 DAY USE 
 
 RETURN TO DESK FROM WHICH BORROWED 
 
 LOAN DEPT. 
 
 This book is due on the last date stamped below, or 
 
 on the date to which renewed. 
 
 Renewed books are subject to immediate recall. 
 
 NOV 8-1966 6 3 
 
 4ftHc3c:W^ 
 
 ..Mr^ ' Bb-^ 
 
 ^ 
 
 \J'u^' 
 
 L.<;^'/^iH 
 
 ij^fur 
 
 M^^m^ 
 
 AUG 21 1984 
 
 CIRCULAT'^M nFPT. 
 
 LD 21A-60ni-7,'66 
 (G4427sl0)476B 
 
 General Library 
 
 University of California 
 
 Berkeley