i — 4> + (j>)=sm {4>-i + 4>),
from which we have at once
tan ()i — <^) = — tan = ^' tan >, ... (3)
i+«i
or
(i+^')tan0 . , (1+^') sin ) cos d)
tan (^1=^ , , , 2 ^ , sm <^i=^ ^ ^.
i — k tan^ 4> A{k, yfei,it
r i+yfei
follows that ^>^i. From (i') it is seen that o<<^<0i, if 4>^-
2
From (2') it is seen that
2
= (i+^i)(i+fe) . . . (i+^„)&i^, . {A)
2
* John Landen, An investigation of a general theorem for finding the length
of an arc of any conic, etc., Phil. Trans. 65 (1775), pp. 283, et. seq.; or Mathe-
matical Memoirs I, p. 32 of John Landen (London, 1780). An article by Cayley
on John Landen is given in the Encyc. Brit., Eleventh Edition, Vol. XVI, p.
153. See also Lagrange, (Euvres, II, p. 253; Legendre, Traite, etc., I, p. 89.
NUMERICAL COMPUTATION 69
where the moduli are decreasing and the amplitudes are
increasing.
It is also seen that
fif) ~—
_ I — Vl— Fr-l IV=1,2, .
I + Vl— yfe2^_i \ ^0 = ^
tan (<^„-0c_i) = V'i-^2___j l-a^n <^„_i. . (i)
It is further evident that F{kn, n) approaches the limit
I dtj)^^, where $ is the limiting value of <^ as w increases.
If <>=-, it follows at once from (i), see also Art. 49, that
2
01= IT, 02 = 27r, . . ., ) =9.8613464
'i-0=36° 0'
20"
<^i = 76° 0'
20"
log i'l = 9.9988771
log tan 4>i =0.6034084
log tan (<^2 — )=^Mki,i), (i)
I+k
i+k
sin {2(t>i — 4>)=k sin ,
where ki>k and (t>i<.
Applying the formula (i) n times, there results
2 2
I+* 1+^1 I+^n-l
or, since
2 ^1 2 k2 .
=—-^=, etc..
1+^ V/fe' i+^i V^i
it is seen that
F{k
,) = k„^ ^'^' ■-■^'^-' F{K^),
where
2V^,-l
h= r^, sin (2 <^,- (/>,-! )
I+Kc-l
= ^„_i sin(/)r-i(j; = i,2, . . . ; ^0 = ^, <^o = <^)-
It follows also that
= r ^'^ rsec0(i.i. = logetan(-+
> V'i-sin2 ^ Jj \4
72 ELLIPTIC INTEGRALS
and
Art. 46. The method of the preceding articles may also
be used to evaluate ^^(30°, 40°), thus
^= .5 log ^ = 9.6989700
1+^ = 1.5 log (1+^) =0.1760913
log V^ = 9.8494850
log 2=0.3010300
colog (1+^) =9.8239087
log ^1 = 9.9744237
^1= .942809 log All = 9.9744237
i+^i = 1.942809 log (i+^i) =0.2884301
log V^ = 9.9872ii8
log 2 =0.3010300
colog (i+*i) =9.7115699
log/b2 = 9-9998ii7
k2= .999567 log ^2 = 9-9998117
1+^2 = 1.999567 log (1+^2) =0.3009359
log V/fez = 9.9999059
log 2 =0.3010300
colog (1+^2) =9.6990641
ki = :
log ki = 0.0000000
log ^ = 9.6989700
log sin <^ = 9.8080675
log sin (2)i - <^) =9-5070375
2)i — <^=i8° 44' 50."o5
201 = 58° 44' 5o."io
01 = 29° 22' 25".o5
NUMERICAL COMPUTATION 73
log Ai = 9.99744237
log sin 01=9.6906403
log sin (2 02 - 4>i) = 9.6650640
202-01 = 27° 32' 43. "08
202 = 56° 54' 68."i3
02 = 28° 27' 34."o6
log ^2 = 9.9998117
log sin 02 = 9.6780866
log sin (2 0.j- 02) =9.6778983
203-02 = 28° 26' 45. "53
203 = 56° 54' i9."59
03 = 28° 27' 9."78
When ^3 = 1, then sin (2 04-0,) = sin 03, or 04 = 03.
••• 04 = 28° 27' 9."78
T = ^4° 13' 34."89
^ = 7+^=59° 13' 34."89
* = S9° 13' 34."89
logiotan$= .225120S
log log tan* = 9.3524156
cologM = o.3622i57 (*see below)
log V^i =9.9872118
log Vife2 = 9.9999059
colog ^^ = 0.1505150
log F(3o°, 40°) =9.8522640
F(30°,4o°)= .711647
Art. 47. Cayley, Elliptic Functions, p. 324, introduced instead
of the standard form of the radical, a new form
Va^cos- (p+b'^ sin- (a>b);
* Division is made by the modulus M to change from the natural to the com-
mon logarithm, where J/=. 43429448.
74 ELLIPTIC INTEGRALS
and he further wrote
F{a, b,
^ cos^ (p+b^ sin^ 4>
E{a, b, 4>) = rVa2 cos^ ,p+b^ sin^ <^.
(i)
(2)
It is clear that
Va^ cos^ +b^ sin^ (j) = aVi — P sin^ (^,
where
*-.-»^,*'=^.
The functions (i) and (2) are consequently -F{k, 0) and
aE{k, ).
Fig. 17.
In the figure let P be a point on the circle, whose centre is
and let Q be any point on the diameter AB.
Further let
QA=a,QB = b, ZAQP = i, ZA0P = 2cj,, £ABP = .
Write ai=\{a+b), bi = Vab, ci=^{a — b).
It follows at once that
OA=OB=OP = ai,OQ = ai-b=^{a-b)=Ci,
QP sin i=ai sin 20,
QP cos 0i=ci+ai cos 20.
NUMERICAL COMPUTATION 75
On the other hand
Qp = ci2+2Ciai cos 20+ai2=^(a2+62)+|(a2-62) cos 2
=|(a2+&2)(cos2 0+sin2 <^) +§ (a2 - ^2) (cos2 0-sin2 )
= a- cos2 (t>+b~ sm2 0.
Therefore it follows that
ai sin 24> , ci+fli cos 2)
sin 01 = ^ , cos 01 = - ^ ■
Va2 cos2 0+^)2 sin2 0' \/a2 (-032 0+^2 sin2 0'
and consequently
2 2 ^ I A " ■ 2 J. '^i^^'^ cos2 0+& sin2 0)2 ,
ai2cos2 0i+6i-sin2 0i= — \ „ , „ . , ^ • • (i)
a-' COS'' 0+0'' sin''
It is seen at once that
h{a — b) sin 2
sin (2 — 0i) =
\/a~ cos2 0+^2 sin2
, . a cos2 + & sin2 r / \
cos (20-0i)= - ; or, from (i),
Va2 cos2 4>+b~ sin2
cos (20 — 0i)=— Vai2 cos2 0i+6i2 sin2 0i .
ai
If in the figure we consider the point P' consecutive to P,
then, PQd(l>i=PP' sin PP'Q = 2ai cos (20-0i)(i0;
or, writing for PQ its value from above, there results
2d4> i
Va^ cos2 0+62 sin2 V'ai2 cos2 0i+6i2 sin2 0i
Integrating, this expression becomes
F{a, b, 0)=§F(ai, 61, 0i),
or
F{k, 0)=i -F{k', 0') = -^ ^(^1. <^i).
2 fli i+«
76 ELLIPTIC INTEGRALS
where
. ^ ^{i+k') sin 2
sin i = , . - •
V I — ^^ sin^ (/)
a? a a^ \a-\-hJ \i+k /
or, yfei = i^^, and ^' = ^ — r^, as given at the end of Art. 42.
i+k i+«i
Art. 48. Cayley derives a similar formula for the integrals
of the second kind as follows, his work being here in places con-
siderably simplified. From the relation of Art. 42, we have
sin {2 4> — 4>i) =ki sin >!, or
sin 24, cos )i — cos 20 sin >i =ki sin i;
it follows that
cos 20= —^1 sin^ 01-1-cos 0i A0i,
and consequently
2 cos^ 0= I —^1 sin- 01 + cos 0iA0i,
2 sin^ = 1+^1 sin^ 0i — cos 0iA0i.
From these two relations it is seen at once that
2 (a2 cos2 + ^2 sin2 0) =a^+b^-{a^-b^)ki sin^ 0i
+ {a^-b^)cos 0iA0i = (a2 + 62)(cos2 0i+sin2 0i)
-ia^-b^)ki sin2 0, + (a2-^)2)cos 0iA0i
= 4(01^ cos2 0i+&r sin2 0i)
— 2bi-+4Ci cos 0iV ai^ cos2 0i+&i2sin2 0i.
Multiply this expression by the differential relation given
above, viz.,
2J0 _ dtj)i
Va^ cos2 0+62 sin2 Var cos^ 0i+6i2 sin^ 0i'
NUMERICAL COMPUTATION 77
and integrating, there results
E{a, b, )='^Eiku 0i) -- —F{ku 00+^ sin 0i,
" 2 flai a
or
£(^, 0)=^(i+y)£(^^, 0i)__A_/7(^j_ 4>i)+ki-k')sm and '_. ,,^ i+^'tan-0 cos^ 0i
d^~^^'^ \i-/fe'tan2 0)2 cos2 '
a positive quantity, it appears that i increases with >. It is
further evident that tan 0i =0 when tan = 00 . It is clear from
(i) that when = o, )= — and £(a„, &„, ^) =a„(/>;
an
* Gauss, Werke, III, pp. 361-404.
NUMERICAL COMPUTATION 79
further if <^=|ir, it is seen that
F{an, b„) = — and £(a„, b„) =- o„, where a„=n.
2a„ 2
The equation F{a, b, <^) =| F{ai, bi, 0i) gives
F{a, b, 4>)=iF{ai, bi, <^i)=^ F{a2, 62, 2)
2*'
2" 2"a„
where the (f>'s are to be calculated from the formula
ai sin 2
sin i =
V a^ cos^ 0+6^ sin^
(Z2 sin 2<^i
sin (j)2=-
\ ai^ cos^ 01 +61^ sin^ (pi
Art. 52. r/je integrals of the second kind. Note that, since
Fia,b, )=E{ai, bi, i)-ai^F{ai, bi, 4>i)
+F{ai, bi, 0i)(ai2-|a2_ijj2)_|_cjsin0i;
or, since ar-^a^-^bi^= -l{a^-b^) = -aici,
the above equation is
E{a, b, 4>)-a^F{a, b, )=E{ai, bi, i)-ai'^F{ai, bi, dn)
— a\CiF{ai, bi, 4>i)+C\ sin (pi.
Observing that, as n increases,
hm [E{an, b„, )-a^F{a, b, 0) = -[2aiCi+4a2C2+SazC3+ ■ ■ ■ ]F{a, b, )
+ci sin 01+C2 sin 02+C3 sin (^3+ • • • ;
or finally
E{a, b, 0) = [a2-2CiCi-4a2C2-8a3C3 - . . . ]F{a, b, )
+ci sin 4>i+C2 sin 02+C3 sin 3+ . . .
80 ELLIPTIC INTEGRALS
In particular, if <#> =|ir, we have Art. 49, <^i =ir, 02 = 2%,
and then
£(a, 6) = [(2^ — 2aiCi — 432^2— . . . ] — .
2 an
It also follows immediately that
F{k, 0)
H — ^sin 01 H — ? sin 02+
a a
or, noting that
aiCi I ,, a2C2 I , (X3C3 I r
= ^^, = fti, = k2, .
02 4 aiCi 4 a2C2 4
Cl ki
c% k2 a\ I
fli i+fe' a i+^i'
C3 ^3 02 I fli 1
02 1+^3' di 1+^2' a i+^i' ■ ■ ■ '
the equation becomes,
E{k, ) = [i-\Hi+\ki+\kik2-¥\kik2h+ . . . )]F{k, 0)
, ^1 . , , k2 ■ ,
■ Sin 01 +;^ -— r Sm 02
I+^l (l+^l)(l+>fe2)'
+ (l+/fel)(lA)(l+/fe3) '^ *' +
Further since
I fe I ^
=, or
I+^i 2Vyfei 1+^1 zV/fei
I ^1 I ^V^i
■= — ^, or -
1+^2 2V/b2' (l+^l)(l+^2) 4Vyfe2'
I _ ^2 I _^v'^i^2
T+kz~2s/Yz °^ (l+*l)(l+^2)(l+^3)~ 8V^ '
NUMERICAL COMPUTATI0\
81
the last line of the above expression may be written
k[lV^ sin 4>i+l\^kik2 sin <^2-Fiv^^i^2*j sin s+ . . . ].
In particular if <^ = §ir, we have
Art. 53. As a numerical example (see Legendre, Traite etc.,
T. I, p. 91) , let a = I, 6 = I "^2 — V3 = cos 75°, and let tan = -J -7=.
\ V3
TO
It follows that k^ = i ^ = sin 75°.
a~
The following table may be at once constructed.
Index
(o)
(i)
(2)
(3)
(4)
I .0000000:
0.6204095
0.5690761
0.56747^4
0.5674713'
0.2588190
0.5087426
0.565S68S
0.5674701
0.5674713
0.3705905
0.0603334
0.0016037
O.OOOOOII
0.965925S, 0.258S190 47 3 31
0.5SS790S
0.1060200
0.002S260
0.0000020
0.S082856 62 36 3
0.9943636 119 55 48
0.9999959240 o o
o. 9990900480 o o
(See Cayley, loc. cit., p. 335.)
The complete integral fi = = 2.768063 . . . and
2 Ui
i^(75°, 47°3'3i")=t'v = o.9226877 • • •
Note that the first integral is three times the second.
It is also seen that
Ui-j-]= aici = .2332532
+ 2a2C2 = .0686686
+4a3C3 = .0036402
+8a4C4 = .0000051
= •3055671
and £1 = 1.0764051
82 ELLIPTIC INTEGRALS
The computation of E{k, ) is found in the next article.
Art. 54. To establish in a somewhat different manner
the results that were given in the preceding article, consider * a
function G{k, (j)) composed of an integral of the first and of an
integral of the second kind, such that
Ju Vl— ^2sin2 (f,
where a and are constants.
Making in this integral the substitutions of Arts. 42 and 48,
namely
d^^i+h d^^ gj^2 ^ = i(i+;fei sin2 4>i-A4>i cos <^i),
it is seen that
G(^, 0) = l±^-i[G(^,, <^i)-i^sin0i], . . . (i)
2
where
.*0=f'^
G{K4>.)=1 ^ii±^ii>Ili^d<^
A4)i
the constants ai and /3i being defined by the relations
We saw in Art. 48 that
, i-Vi-F ^ ^ , — -
Ki = . , tan {i — ,
I + V I — ^2
where ^1 <^ and 4>i> 4>.
It follows directly from (i) that
22 2
iTi+^i . i+^i 1+^2, . _, ,
— /3 sin <)!)iH jSi sm 4)2+
2L 2 22
i+^i 1+^2 i+^"a • . 1
22 2 J
* See also Legendre, Traile, etc., I, p. 108.
NUMERICAL COMPUTATION
83
where
k\k2
and
/3p = /3
a„=a+|/3(l
^1 k\k2
k\k2
2 2^ 2"-
Since /3„ becomes o with kn, it is seen that
lim G{h
X*
a„ d(i)=a„4)n.
From Art. 43 we had
i+^i 1+^2 I+^n
n=F{k, ),
and, see Art. 42,
2 k ' 2 k\ '
It follows that the above formula becomes
G(^,^)=f(^,<^)[.+^i+^+^^+'-^^+ . . .)_
— T sin <^iH 5— sin <^2H n — sm z+ .
k\ 2 2^ 2^^
If in this formula we put a = i, /3= — A^, it becomes
1^2 , k-[k2kz
£(^,.^) = F(^,<^)[i-f(i+^^+^^
"^'^i_:_ ^ I ^kik2,
2
+jfep-^sin «/>i+ ".;"" sin <)2-
5 — sm 03+ .
where
/Cp "^
1 — Vi— A;Vi
i + Vi-)feVi
and
tan (0p— <^p-i) = v^i-'feVi tan <^p-i.
84 ELLIPTIC INTEGRALS
These results verify those of Art. 52.
With Legendre, Fonct. Ellip., T. I., p. 114, we may find
E{k, 4>) where ^ = sin 75° and tan ) = ^/— ^.
Using the results of Art. 53 it is seen that
sin (^1 = .3290186
sin 02= .0522872
sm 4>2=— .C013888
kv kik2k.:ki .
Sin (j>i = .0000010
16
Writing
sum = .3799180
24 8 16 '
it is found that Z, = . 3888658 . . .
In Art. 53 it was seen that F{k, 4>) = .9226877 . . .
It follows that E{k,(j))=F{k,<^))L+.s^ggI8o
0.7387196 . . .
Further since
E
there follows
'■t)-i'-t)'-
£1=1.0764049 . . .
Art. 55. Inverse order of transformation. If the modulus k
is nearer unity than zero, the following method is preferable.
The equation (i) of the preceding article may be written
G{ki, 4>i)=^G{k, 0)+|i sin <^i, since fi = ^.
NUMERICAL COMPUTATION 85
If in this formula the suffixes be interchanged, then
where
now
2 B
G{k, (i>)=——-G{ki, 0i)+-sin0,
i+k k
.2^ ^
ki= r, sin (201 — 0) =^ sin (^,
i+«
ki>k, <^i<).
The continued repetition of (2) gives
G{k, (t>)=^ sin 4>+-^ sin 0iH — -=^^2 sin 4,2
k Vk Vk
-f — ;=— |33Sin<^jH 7= /3„-i sin 0„-i
Vk
where
kki . . . kp-i
and
^\ ki k\k2 ' ' ' k\k2 ■ ■ ■ kp-i
Since K approaches unity (rapidly) as n increases,
^ r*f.a„+/3„sin2 ,
limG(*„, ^^
= (a„ + ^n) loge tan(- + Y) -^n S^ <#>»•
In Art. 45 it was shown that
^ k^^bEI^lo, tan(^+^) =F{k, 0)
86 ELLIPTIC INTEGRALS
We may consequently write the above formula
G{k, )-F{k, )[a-^-(^z+^+-^+ . .
^1^2 • • ■ ^n-1
,2 2^
Vk Vkki V/feyfeiyfe2
+- sin^H — ^sin<^iH — ^==sm4>2-i — , ^_ , sin 3+ ■ ■ .
^sin0„-i —
Vkkl . . . k„-2 Vkki . . . k
Writing a = i , /3 = — yfe^ in this formula, it becomes
E{k, 4,)=F{k, 4>)^i+k(i+^+~+ . . .
2"-^ 2" y
klk2 . . ■ kn-1 ^1^2 ■ • • kn-\l .
/ 2 2^
— ^|sm(/)H — -=sin<>iH — p=sm<^2+ ■ • •
\ Vk Vkkl
= sin„-!-
= sin<^„),
V ^^1^2 . • • ^n-2 vkkl ■
where
kp = — — ^ — and sin {2p — p~i)=kp-i sin p-i.
i+kp-i
Taking the example of the preceding article, and using the
values given in Art. 53, it is seen that
— k sin = —0.7071070
— 2 Vk sin 01 = — 1 .4 146540
Vk
+4—= sin 4)2 = 2.8293085
Vki
F{k, 0) = .9226877
and
F(^, (^) 1+^-— = 0.0311720
E{k, )= 0.7387195 . . .
NUMERICAL COMPUTATION 87
Art. 56. Two of the principal problems that appear in
practice will now be given.
Problem i. When u and k are given, calculate the values oj snu, cnu,
dn u.
1. Computation of snu. In the Table II, p. 96, is found an imme-
diate answer to the problem.
For when u and ^ = sin e are known, the value 4> may be found in
the table and then sn u from the formula sn !< = sin (i>.
If, for example, ^ = 5 = sin d, and m = .47S5i, it is seen that for 9=30°,
M = .47SSi, we have =2-]°, and sin 0=. 45399 = 5;! u.
2. The computation of en u and dn u are had from the formulas
cnu= ±V (i— in u){i-\-snu),
dnu= :ii^{i—ksnu){i+ksnu).
Problem 2. Having give^i the elliptic Junction, calculate the argument.
1. If snu is known, find u. Table II furnishes the solution. Sup-
pose that a is the given value of snu, and suppose that k = s\n B is also
known. Hence, since snu = sva.^=a, we may determine <(>. With e
and known, we find the value of u from the table. Denote this value
by Mo- From the relation snu = sn Uo, we have (Art. 21),
M = Mo +4tnK ■\-2m'iK'.
Further in the formula (Art. 12).
sn u= —sn{u-]-2K),
substitute u=—uo, and then we have —snuo=—sn{2K—Uo), so that
u may also have the form
« = 2K—Uo+4mK+ 2m'iK'.
2. li en u and dn u are given, snu and then u may be found as above.
CHAPTER V
MISCELLANEOUS EXAMPLES AND PROBLEMS
I. The rectification of the lemniscate. The equation of the curve is
(y2+i-2)2+a2(/-a:2) = o;
or, writing x=r cos 6, y = r sin 9, the equation becomes
)-2 = (j2 cos 2S.
From the expression ds''- = dr'^-\-r-d9-, the differential of arc is
^ a p( T- ^
Jo Vi-jsin^e V2J0 Vi-isin^,^ V2 VVI' j
which may be calculated at once from the tables when a and 9 (or 0)
are given. A quadrant of the lemniscate is
Jo Vi-2sin2e "^zjo Vi-isin2 V^ ^^2
2. r/ze rectification of the ellipse.
Let the equation b
From the integral
X^ -y2
Let the equation be — \-— = i,a>b
a^ b^
"IM.
dx,
a'^ — f
we have, by writing k^= — - — , x=at,
a'
-kV)dt
^ P (i-kV)d
Jo V(l-/2)(i_
kV)
Fig. 18.
Finally writing / = sin (see Art. 3) and that is x=a sin 0, we have
j= I Ad(t>=aE{).
Jo
SS
MISCELLANEOUS EXAMPLES AND PROBLEMS
89
Here k is the numerical eccentricity of the ellipse. The angle 4, = C0Y =
90— CO^, where in astronomy the angle COA is known as the eccentric
anomaly of the point P Writing = Tr/2, it is seen that the quadrant
of the ellipse is aE, where E is the complete integral of the second kind.
If the equation of the ellipse is taken in the form
*=osin0, y = 6coS(^,
it follows at once that
, ds'^ = a-{i — k'^sm'^ (t>)d'^, or s=aE{).
3. The major and minor axes of an ellipse are 100 and 50 centimeters
respectively. Find the length of the arc between the points (o, 25) and
(48, 7). Find also the length of the arc between the points (48, 7) and
(50, o). Determine the length of its quadrant.
4. If X denotes the latitude of a point P on the earth's surface, the
equation of the ellipse through this point as indicated in the figure, may
be written in the form
a cos X
x = -
It follows at once that
tf2(l-t.2
y=
a(i—e^) sin X
Vi — e'^sia^ X
(i-
ds''=dx^+dy-' =
so that
s=aii-e') I
Jo (i-
e^ sin- X)''
d\
Fig.
19.
-e- sin^ X)
This integral may be at once
evaluated by the third formula
in Art. 41.
Compute the lengths of arc of the ellipse between 10° and 11° and
between 79° and 80° where a =6378278 meters and €- = 0.0067686.
Compare these distances with the length of an arc that subtends i" upon
a circle with radius=6378278 meters.
5. Plot the curves, the elastic curves, which are defined through the
differential equation
, y-dv
d^=± ; ,
Va"— y*
for the values a= i, 2, 4, 9.
6. The axes of two right cylinders of radii a and b respectively (a > b)
intersect at right angles. Find the volume common to both.
Let the z-axis be that of the larger cylinder and the y-axis that of
the smaller, so that the equations of the cyhnders are
x'^+y^=a- and x^+z^=b^ respectively.
The volume in question is
V
-j:
Va^-x^ Vb'^-x'^dx.
90 ELLIPTIC INTEGRALS
Writing t = sn~H j-,-)> (see formula 5a, Art. 23), then x=bsnt, b^—x^=
hhnH, a.'^—x^=aHnH, d = b cnidnldt.
It follows that
F=8a62 I i- ° „ snH-\-—snH\di. (SeeByerly,/«^ Cai., i902,p. 276.)
Joting (see sixth formula of Art. 41, and (ii) of Art. 48) that
Jr-K I (-K I
k" Jo '
Noting (see sixth formula of Art. 41, and (ii) of Art. 48) that
it follows at once that
Compute V when a = 60 and 6=12 centimeters respectively; also
find the volume common to both when the shortest distance between
the axes is 8 centimeters.
7. The differential equation of motion of the simple pendulum is
dh dy
dfi^~^Js'
or multipljnng by -3- and mtegratmg,
dt
|y=-2gy+C.
If the pendulum bob starts from the lowest point of its circular path with
the initial velocity that would be acquired by a particle falling freely
in a vacuum through the distance yo, so that iio^=2gyo (Byerly, loc. cit.,
p. 215), it is seen that this is the value of C, and consequently
Further taking the starting-point as the origin (see figure) the equation
of the circular path is x^+y'^—2ay=o, so that
fds\ ' a' l^y\ ^
\dt/ 2ay—y'^\dt
and consequently
dy
t =
'^2gJo ^/(yo-y)(2ay-/)'
which is the time required to reach that point of the path whose ordinate
is y.
ting k^=
2a ya
yo y
Writing fe^= — and sin^ =—, this integral becomes at once
/ ^. = \-Fik,,p).
Vi-Ai^sin^* S
MISCELLANEOUS EXAMPLES AND PROBLEMS
91
Let OC=CA = ahe the length of the pendulum. Let A be the highest
point reached by it in the oscillation so that the ordinate of A is y^. Let
the angle ACO be a, and let e be the angle PCO, where P is the point
reached at the expiration of the time t.
It is seen that
yo
— =1 — COSc,
a
so that
iV
V2=x/K:-c<
and similarly,
V — =sm-.
It follows also that
V-^ sin -
yo . a
sin -
2
When e=a, sin 0=1, or = -, and consequently, the time of a half-oscilla-
, ■ ■ .P
lation IS \- i
F(sin-,^
2 2
Show by Table I that when = 36°, the time of oscillation is 1.0253 • • ■
la
times greater than that given by the approximate formula / = "V - •"-.
g
The following problems taken from Byerly's Calculus are instructive:
8. A pendulum swings through an angle of 180°; required, the time
of oscillation. Ans. 3.708V-.
9. The time of vibration of a pendulum swinging in an arc of 72° is
observed to be 2 seconds; how long does it take it to fall through an
arc of s", beginning at a point 20° from the highest point of the arc of
swing? Ans. 0.095 • ■ second.
10. A pendulum for which
[a. .
vibrates through an arc of 180°;
through what arc does it rise in the first half second after it has passed
its lowest point? In the first | of a second? A7ts. 69"; 20° 6'.
1 1 . Show that a pendulum which beats seconds when swinging through
an angle of 6°, will lose 11 to 12 seconds a day if made to swing through
8° and 26 seconds a day if made to swing through 10°.
(Simpson's Fluxions, § 464.)
CHAPTER VI
FIVE-PLACE TABLES
The following tables of integrals are given in Levy's Theorie
des fonctions elliptiques. As stated by Professor Levy, he was
assisted by Professor G. Humbert in compiling these tables from
the ten-place tables that are found in the second volume of
Legendre's Treatise.
Table I gives values of the integrals
K= P'— =ii=== and E= C'd4>Vi-sm^ 6 sin^ .
Jo Vi — sin^ d sin^ Ja
For example, if 6 = 78° 30', then iT = 3.01918 and £ = 1.05024.
Table II gives values of the integral
d
F{k,4>) =
i v^
- sin^ d sin^ >
For example, if ^ = 65° and <^ = 8i°, then F(k,) = 1.94377.
Table III gives values of the integral
E{k, V I -siii^ 9sin2 0.
For example, if 0=40° and <^ = 34°, then E{k, <>) =0.57972.
92
FIVE-PLACE TABLES
93
I. -THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND
SECOND KINDS
8
K
E
6
K
E
6
K
E
o°
1.57080
1 . 5 7080
50°
1-93558
1-30554
82° 0'
3-36987
I .02784
I
092
068
51
5386
29628
12
9457
670
2
127
032
52
7288
28695
24
3-41994
SS8
3
187
1.56972
53
9267
27757
36
4601
447
4
271
888
54
2.01327
2681S
48
7282
338
5
379
781
55
3472
25868
83
3 . 50042
231
6
511
650
S6
5706
24918
12
2884
126
7
668
495
57
8036
23966
24
5814
023
8
849
296
58
2 . 10466
23013
36
8837
1921
9
I ■ 58034
114
59
3002
22059
48
3-61959
821
10
284
1.55880
60
2.15652
2 1 106
84
3-65186
I. 01 724
II
539
640
61
8421
20154
12
8525
628
12
820
368
62
2.21319
19205
24
3 71984
534
13
I 59125
073
63
4355
18259
36
5572
443
M
457
I -54755
64
7538
17318
48
9298
354
IS
814
415
65
2.30879
16383
85
3-83174
266
i6
I 60IQ8
052
66
4390
15455
12
7211
181
17
608
1-53667
67
8087
14535
^■i
3-91423
099
i8
1.6104s
260
68
2.41984
13624
36
5827
018
19
510
t -52831
69
6100
12725
48
4.00437
0940
20
1.62003
380
70° 0'
2 ■ 50455
11838
86
5276
86s
21
523
I. 5 1908
30
2729
"399
12
4.10366
792
22
I 63073
415
71
5073
10964
24
5736
721
23
6.S2
I . 50901
30
749°
10533
36
4.21416
653
24
1.64260
366
72
9982
106
48
7444
588
25
900
1.49811
30
2-62555
09683
87
4-33865
526
26
1-65570
237
73
5214
265
12
40733
466
27
6272
1.48643
30
7962
8851
24
8115
410
2S
7006
029
74
2 . 70807
443
36
56190
356
29
7773
1-47397
30
3752
039
48
64765
306
3°
I -68375
I .46746
75
6806
7641
88
74272
258
31
041 1
077
30
9975
248
12
84785
215
32
I 70284
I 45391
76
2.83267
6861
24
96542
174
33
1192
44687
30
6691
480
36
5-09876
137
34
2139
43966
77
2.90256
106
48
25274
J04
3S
3125
229
30
3974
5738
89
43491
07s
36
4150
42476
78
7857
378
6
54020
062
37
5217
41707
30
3,01918
024
12
65792
050
38
6326
40924
79
6173
4679
18
79140
049
39
7479
126
30
3 . 10640
4341
24
94550
030
40
1.78677
I -39314
80
5339
oil
30
6.12778
021
41
9922
38489
12
7288
3882
36
35038
014
42
1.81216
37650
24
9280
754
42
63854
008
43
2560
36800
36
3-21317
628
48
7-04398
004
44
3957
35938
48
3400
503
54
737"
on
45
5407
35064
81
5530
379
90
00
000
46
6915
34x81
12
77"
257
47
8481
33287
24
9945
126
48
I. 90108
32384
36
332234
017
49
1800
31473
48
4580
2900
94 ELLIPTIC INTEGRALS
II.— ELLIPTIC INTEGRALS OF THE FIRST KIND
e
0°
5°
10°
15°
20°
25°
30°
3S"
40°
45°
~I~
0.0174s
0.01745
0.0174s
0.0174s
0.01745
0.01745
0.01745
0.01745
0.0174s
0.01745
2
03491
03491
03491
03491
03491
03491
03491
03491
03491
03491
3
05236
05236
05236
05236
05236
05236
05237
05237
05237
05237
4
06981
06981
06981
06982
06982
06982
06983
06983
06984
06984
5
08727
08727
08727
08727
08728
08729
08729
0S730
08731
08732
6
10472
10472
10473
10473
10474
10475
10477
10478
10480
10482
7
12217
12218
12218
12219
12221
12223
12225
12227
12230
12233
8
13963
13963
13964
13966
13968
13971
13974
13978
13981
13985
9
15708
15708
15710
15712
15715
15719
15724
15729
IS735
15740
lO
17453
17454
17456
17459
17464
17469
17475
17482
17490
17498
II
19199
19200
19202
19206
19212
19220
1922S
19237
19247
19258
12
20944
20945
20949
20954
20962
20971
20982
20994
21007
21021
13
22689
22691
22695
22702
22712
22724
22738
22753
22770
22787
14
24435
24436
24442
24451
24463
24478
24495
24514
24535
24556
15
26180
26182
26189
26200
2621S
26233
26254
26278
26303
26330
i6
27925
27928
27936
27949
27967
27989
28015
28044
28075
28107
17
29671
29674
29684
29699
29721
29748
29779
29813
29850
29889
i8
31416
31420
31431
31450
31475
31507
31544
31585
31629
31675
19
33161
33166
33179
33201
33231
33268
33312
33360
33412
33466
20
34907
34912
34927
34953
34988
35031
35082
35138
35199
35262
21
36652
36658
36676
36706
36746
36796
36855
36920
36990
37063
22
38397
3S404
38425
38459
38505
38563
38630
38705
387S6
38871
=3
40143
40151
40174
40213
40266
40331
40408
40494
40587
40683
24
41888
41897
41924
41968
42027
42102
42189
42287
42392
42503
25
43633
43643
43674
43723
43791
43875
43973
44084
44203
44328
26
45379
45390
45424
45479
45555
45650
45761
45885
46020
46161
27
47124
47137
47174
47236
47321
47427
47551
47690
47841
48000
28
48869
48883
48925
48994
49089
49207
49345
49500
49669
49846
29
5061 s
50630
50677
50753
50858
50988
51142
51315
51503
S1700
30
52360
52377
52428
52513
52628
52773
52943
53134
53343
53562
31
54105
54124
54181
54273
54401
54560
54747
S4959
55189
55432
32
55851
55871
55933
5603 s
56175
56349
56555
56788
57042
57310
33
57596
57619
57686
57797
579SO
58141
58367
58623
58902
59197
34
S934I
59366
59439
59561
59727
59936
60183
60463
60769
61093
35
61087
61113
61193
61325
61506
61734
62003
62308
62643
62998
36
62832
62861
62948
63090
63287
63534
63827
64159
64524
64912
37
64577
64609
64702
64857
65070
65337
65655
66016
66413
66836
38
66323
66356
66457
66624
66854
67144
67487
67879
68309
68769
39
68068
68104
68213
68393
68641
68953
69324
69747
70214
70713
40
69813
69852
69969
70162
70429
70765
71165
71622
72126
72667
41
71558
71600
71726
71933
72219
72580
73010
73502
74047
74632
42
73304
73349
73483
73704
74011
74398
74860
75389
7,5976
76608
43
75049
75097
75240
75477
75805
76219
76714
77282
77914
78594
44
76794
76846
76998
77251
77600
78043
78573
79182
79860
80592
45
0.78540
0.78594
0.78756
0.79025 0.79398
0.79871
0.80437
0.81088
0.81815
0.82602
FIVE-PLACE TABLES
95
n.— ELLIPTIC INTEGRALS OF THE FIRST KIND
*
e
50°
55°
60°
65°
70°
75°
80°
85°
90°
T°
0. 01 745
0.0174s
0.01745
0.01745
0.01745
3.01745
0.01745
0.01745
0.01745
2
03491
03491
03491
03491
03491
03491
03491
03491
03491
3
05237
05232
05238
05238
05238
05238
05238
05238
05238
4
0698s
06985
06986
06986
06986
06987
06987
06987
06987
S
08733
08734
0873s
08736
08736
08737
08737
08738
08738
6
10483
10485
10486
104S8
10489
10490
1 049 1
10491
10491
7
1^235
12238
12240
12242
12244
12246
12247
12248
12248
8
139S9
13993
13997
14000
14003
14005
14007
14008
14008
9
15746
15751
15757
15761
15765
15769
15771
15772
15773
lO
17505
17513
17520
17526
17532
17536
17540
17542
17543
II
19268
19278
19288
19296
19304
19310
19314
19317
19318
12
21034
21047
21059
21071
21080
21088
21094
21098
21099
13
22804
22821
22836
22851
22863
22873
22880
22885
22886
14
24578
24599
24618
24636
24652
24664
24674
24680
24681
IS
26356
26382
26406
26428
26448
26463
26475
26482
26484
i6
28139
28171
28200
28227
282S1
28270
28284
28293
2829s
17
29927
20965
30001
30034
30062
30085
30102
30112
30116
i8
31721
31766
31809
31848
31881
31909
31929
31942
31946
19
33520
33574
33624
33670
33710
33742
33766
33781
33786
20
35326
35388
35447
3S50I
35548
35586
35615
35632
35638
21
37137
37210
37279
37342
37396
37441
37474
37494
37501
22
38956
39040
39119
39192
39255
39307
39346
39369
39377
23
40782
40878
40969
41053
41126
41186
41230
41257
41266
24
42614
42724
42829
42925
43008
43077
43128
43159
43169
25
44455
44580
44699
44808
44904
44982
45040
4507s
45088
26
46304
46445
46580
46704
46812
46901
46967
47008
47021
27
48161
48320
48472
48612
48735
4883 s
48910
48956
48972
28
50027
50206
50377
50534
50672
50785
50870
50922
50939
29
51902
52102
52293
5247c
52624
52752
52847
52905
52925
30
53787
54009
54223
54420
54593
54736
54843
54908
S493I
31
55681
55928
56166
S6386
56579
56739
56858
56931
55956
32
33
34
3S
57586
57860
58123
58367
58582
58760
58893
5897s
59003
59501
59803
60095
60365
60604
60802
60950
61042
61073
61427
61760
62082
62381
62646
62865
63029
63131
63166
63364
63730
64085
64415
64707
64950
65132
6524s
65284
36
37
65313
65715
66104
66468
66790
67058
67260
67385
67428
67273
67713
68141
68540
68895
69131
69414
69552
69599
?8
69246
69727
70195
70633
71023
71340
7,1594
71747
71799
39
40
71232
73231
71756
73801
72267
74358
72746
74882
73175
75352
73533
75745
73804
76043
73972
76228
74029
76291
41
42
43
44
4S
75243
77269
79308
81362
0.83431
75862
77940
8003 s
82149
0.84281
76469
78600
80752
82926
0.85122
77041
79224
81432
83665
0.85925
77555
79786
82045
84333
0.86653
77987
80258
82562
84898
0.87270
78313
80617
82954
85329
0.87741
78517
80841
83200
85598
0.88037
78586
80917
83284
85690
0.88137
96
ELLIPTIC INTEGRALS
II.— ELLIPTIC INTEGRALS OF THE FIRST KIND
e
0°
5"
10°
15"
20°
25°
30°
3S°
40°
45°
46°
0.8028s
0.80343
0.80515
0.80801
0.81198
0.81701
0.82305
0.83001
0.83779
0.84623
47
82030
82092
82275
82578
82999
83535
84178
84920
85752
86656
48
83776
83841
84035
84356
84803
85371
86055
86846
87734
88701
49
85521
85590
85795
8613s
86609
87211
87937
88779
89725
90759
50
87266
87339
87556
87915
88416
89054
89825
90719
91725
92829
SI
89012
89088
89317
89697
90226
90901
91716
92665
93735
94912
52
90757
90838
91078
91479
92037
92750
93613
946 1 8
9S7SS
97007
S3
92502
92587
92841
93262
93850
94603
95514
96578
97784
0.991 1 5
54
94248
94337
94603
95047
95666
96458
97420
0-98545
0.99822
101237
SS
95993
96086
96366
96832
97483
0.98317
0.99331
1.00519
1.01871
03371
S6
97738
97836
98130
0.98618
0.99302
1. 001 79
1. 01 24 7
02499
03928
05519
57
0.99484
0.99586
0.99894
1 .00406
1. 01 1 23
0204^
03167
04487
05996
07680
58
1.01229
1.01336
1.0165?
02194
02946
03912
05092
06481
08073
09854
59
02974
03086
03423
03984
04770
05783
07021
08482
10159
12042
60
04720
04837
05188
05774
06597
07657
08955
10490
12256
14243
61
06465
06587
06954
07566
08425
09534
10894
12504
14361
16457
62
08210
08338
08720
09358
10255
11414
12837
14525
16476
18685
63
09956
10088
10486
11151
12087
13296
14784
16552
18601
20926
64
1 1 701
1 1 839
12253
12945
13920
15182
16735
18586
2073s
23180
6S
13446
13590
14020
14740
I57S5
17070
18691
20626
22877
25447
66
15192
15340
15787
16536
17592
1 896 1
20651
22672
25029
27727
67
16937
17091
17555
18333
19430
20854
2261S
24724
27190
30020
68
18682
18842
19324
20130
21269
22750
24583
26782
29359
32325
69
20428
20593
21092
21928
23110
24648
26555
28846
31537
34642
70
22173
22345
22861
23727
24953
26548
28530
30915
33723
36972
71
23918
24096
24630
25527
26796
28451
30509
32990
35917
393^3
72
25664
25847
26400
27328
28641
30356
32491
35070
38118
41666
73
27409
27599
28169
29129
30488
32263
34477
37155
40328
44030
74
29154
29350
29939
30930
32335
34172
36466
39244
42544
46404
75
30900
31102
31710
32733
34184
36083
38457
41339
44767
48788
76
32645
32853
33480
34535
36034
37996
40452
43437
46997
51183
77
34390
34605
35251
36339
37884
39911
42449
4S540
49232
53586
78
36136
36356
37022
38143
39736
41827
44449
47647
51474
55999
79
37881
38108
38793
39947
41588
43744
46451
49757
53721
58419
80
39626
39860
40565
41752
43442
45663
48455
51870
55973
60848
81
41372
41612
42336
43S57
45296
47583
50462
53987
58230
63283
82
43117
43364
44108
45362
47150
49504
52470
56106
60491
65725
83
44862
45115
45879
47168
49005
51426
54479
58228
62756
68172
84
46608
46867
47651
48974
50861
53350
56490
60352
65024
7062^
8S
48353
48619
49423
50781
52717
55273
58503
62478
67295
73082
86
50098
50371
51195
52587
54574
57198
60516
64605
69569
75542
87
51844
52123
52968
54394
56431
59123
62530
66734
71844
78006
88
53589
53875
54740
56200
58288
61048
64545
68864
74121
80472
89
55334
55627
56512
58007
60145
62974
66560
70994
76399
82939
90
1.57080
1-57379
1.58284
1.59814
1.62003
1 .64900
1,68575
1-73125
1.78677
1.85407
FIVE-PLACE TABLES
97
U
.—ELLIPTIC INTEGRALS OF THE FIRST KIND
*
50°
55°
60°
65°
a
70°
75°
80°
85°
90°
46°
0.85515
0.86431
0.87342
0.88213
. 89005
0.89678
0.90193
0.90517
0.90628
47
87614
88601
89585
90529
91390
92224
92687
93042
93163
48
89729
90791
91853
92875
93811
94610
95226
95614
95747
49
91860
93001
94146
95252
96267
97139
0.97810
0.98235
0.98381
SO
94008
95232
96465
0.97660
0.98762
0.99711
I . 00444
1 . 00909
I .01068
SI
96171
97484
0.98811
I .00102
1.01297
I .02329
03129
03638
03812
S2
0.98352
0.99759
1.01185
02578
03872
0499s
05868
06425
06616
S3
1.00550
1.02055
03587
05089
06491
07711
08665
09274
09483
54
02765
04374
06018
07637
09155
10481
11521
12188
12418
SS
04998
06716
08479
10223
11865
13307
14442
15171
15423
S6
07248
09082
10971
12848
14624
16190
17430
18229
18505
57
09517
11472
13494
15513
17433
19136
20488
21364
21667
S8
1 1803
13886
16050
18220
20295
22145
23623
24582
24916
59
14108
16325
18638
20970
23212
25223
26837
27890
28257
60
16432
18788
21254
23764
26186
28371
30135
31292
31696
61
18773
21277
23916
26604
29219
31594
33524
34795
35240
62
21134
23792
26606
29490
32314
34897
37008
38407
38899
63
23513
26332
29332
32425
35473
38281
40594
42135
42679
64
25910
28898
32094
3S409
38699
41753
44288
45989
46591
6S
28326
31491
34893
38443
41994
45316
48098
49977
50645
66
30760
34109
37728
41529
45360
48976
52031
541 1 2
54855
67
33212
36753
40600
44668
48800
52738
56096
58404
59232
68
35683
39423
43510
47860
52317
56606
60303
62868
63794
69
38171
42119
46457
51107
55913
60586
64661
67518
68557
70
40677
44840
49441
S44IO
S9S9I
64684
69181
72372
73542
71
43200
47587
52463
57768
63352
68905
73877
77450
78771
72
45739
50359
55522
61182
67198
73256
78759
82774
84273
73
48296
53155
58618
64653
71132
77743
83844
88370
90079
74
50867
55974
61750
68180
75155
82371
89146
1.94267
1.96226
7S
53455
58817
64918
71763
79269
87145
1.94682
2.00499
2.02759
76
56056
61682
68120
75401
83473
92073
2 . 00470
07106
09732
77
58672
64569
71356
79094
87768
1.97157
06529
14136
17212
78
61302
67476
74625
82840
92154
2.02403
12878
21644
25280
79
63943
70403
77924
86637
1.96630
07813
19538
29694
34040
80
66597
73347
81253
90484
2.01193
13390
26527
38365
43625
81
69261
76309
84609
94377
05840
19131
33866
47748
54209
82
7193s
79286
87991
1.98313
10568
25035
41569
57954
66031
83
74618
82278
91395
2.02290
15371
31097
49648
69109
79422
84
8S
77309
85281
94821
06303
20244
37309
58105
81362
2 .94870
80006
88296
1.98264
10348
25178
43658
66935
2.94869
3 13130
86
82710
91320
2.01723
14421
30166
50129
76116
3.09782
35467
87
88
85418
94351
05194
18515
35198
56703
85612
26198
3 64253
88129
1.97388
08674
22627
40265
63357
2.95366
44116
4.04813
89
90843
2.00429
12161
26750
45354
70068
3 05304
63279
4.7413s
90
1.93558
2.03472
2.15652
2.30879
2.50455
2 . 76806
3-15339
3.83174
00
98
ELLIPTIC INTEGRALS
III— ELLIPTIC INTEGRALS OF THE SECOND KIND
*
e
0°
5°
10°
15"
20°
25°
30°
35°
40°
45"
I^
0.01745
0.01745
0.01745
0.01745
0.01745
0.01745
0.01745
0.01745
0.01745
0.0174s
2
03491
03491
03491
03491
03491
03491
03490
03490
03490
03490
3
05236
05236
05236
05236
05236
05236
0523s
05235
05235
05235
4
06981
06981
06981
06981
06981
06980
06980
06979
06979
06978
5
08727
08727
08726
08726
08725
08725
08744
08723
08722
08721
6
10472
10472
1047 1
10471
10470
10469
10467
10466
10464
10462
7
12217
12217
12216
12215
12214
12212
12210
12207
12205
12202
8
13963
13962
13961
13960
13957
13955
13951
13948
13944
13940
9
15708
15707
15706
15704
15700
15696
15692
15687
15681
15676
10
17453
17453
17451
17447
17443
17438
1 743 1
17427
17417
17409
II
19199
19198
19195
19191
19185
19178
19169
19160
19150
19140
12
20944
20943
20939
20934
20926
20917
20906
20894
20881
20868
13
22689
22688
22683
22676
22667
22655
22641
22626
22609
22593
14
24435
24433
24427
24419
24406
24392
24374
24355
24335
24314
15
26180
26178
26171
26160
26145
26127
26106
26083
26058
26032
i6
27925
27923
27914
27901
27883
27S61
27836
27807
27777
27746
17
29671
29667
29658
29642
29620
29594
29563
29529
29493
294SS
i8
31416
31412
31401
31382
31357
31325
31289
31248
31205
31161
19
33161
33157
33143
33121
33092
33055
33012
32965
32914
32862
20
34907
34901
34886
34860
34825
34783
34733
34678
34619
34558
21
36652
36646
36628
36598
36558
36509
36451
36387
36319
36249
22
38397
38390
38370
38336
38290
38233
38167
38094
38015
37934
23
40143
40135
40111
40073
40020
39955
39880
39796
39707
39614
24
41888
41879
41852
41809
41749
41676
41590
41496
41394
41289
25
43633
43623
43593
43544
43477
43394
43298
43191
43076
42958
26
45379
45367
45333
45278
45203
45110
45002
44882
447S3
44620
27
47124
47111
47074
47012
46928
46824
46703
46569
46425
46276
28
48869
48855
48813
48745
48651
48536
48402
48252
48092
47926
29
50615
50599
50553
50477
50373
50245
50097
49931
49753
49569
30
52360
52343
52292
52208
52094
51953
51788
SI 605
51409
5 1 205
31
54105
54086
54030
53938
53813
53657
53476
S3275
S3059
52834
32
55851
55830
55768
55667
55530
55360
55161
54940
54703
54456
33
57596
57573
57506
57396
57245
57059
56842
56600
56341
56070
34
59341
59317
59243
59123
58959
58756
58520
58256
57972
57677
35
61087
61060
60980
60850
60672
60451
60194
59907
59598
59276
36
62832
62803
62716
62575
62382
62143
61864
61552
61217
60868
37
64577
64546
64452
64300
64091
63832
63530
63193
62830
62451
38
66323
66289
66188
66023
65798
65519
65193
64828
64436
64027
39
68068
68031
67923
67746
67503
67203
66851
66459
66035
65594
40
69813
69774
69658
69467
69207
68884
68506
68084
67628
67153
41
71558
71517
71392
71188
70909
70562
70157
69703
69214
68703
42
73304
73259
73126
72907
72609
72238
71804
71318
70793
70245
43
75049
75001
74859
74626
74307
73910
73446
72927
72365
71778
44
76794
76744
76592
76343
76003
75580
75085
74530
73931
73303
45
0.78540
0.78486
0.78324
0.78059
0.77697
0.77247
0.76720
0.76128
0.75489
0.74819
FIVE-PLACE TABLES gg
III.-ELLIPTIC INTEGRALS OF THE SECOND KIND
e
So°
55°
60°
65°
70°
75°
80°
85°
90°
X
°o.oi74
50,0174
50.0174
; 0.01745 0.0174s 0.01745 0.0174^ 0.0174'; o.oi7^c
2
0349<
3 0349(
3 0349(
3 034g(
3 03490 03490 03490 03490 03490
3
4
S
0523.
0697^
0872c
) 0523.
i o697f
) 0871C
^ 0523.
i 0697
) 0871!
i 0523^
7 0697
i 0871S
t 05234 05234 05234 05234 05234
! 06976 06976 06976 06976 06976
! 08717 08716 08716 08716 08716
6
10461
1045?
) 1045S
i I045f
I045i
1045,
i 104S3
1045;
10453
7
1219s
1219-
1219;
12192
I2igc
1218c
) 12188
1218;
12187
13917
15643
17365
8
I393t
13932
13925
13925
13923
1392c
> 13919
1391J
9
IS67C
1566 =
15 66c
15655
15651
1564?
' 15645
15644
lo
17401
17394
17387
17381
17375
17371
17367
17365
II
1913c
1912c
19110
19102
1909s
19085
19084
19082
19081
20791
12
208SS
20842
20830
20819
20809
20801
20796
20792
13
22576
22SS9
22544
22530
225x8
22508
22501
22497
22495
14
24293
24272
24253
24236
24221
24209
24200
24194
24192
IS
26006
25981
25957
25936
25917
25902
25891
25884
25882
i6
27714
27684
2765s
27629
27606
27588
27575
27567
27564
17
29418
29381
29347
293 1 5
29288
29267
29250
29241
29237
i8
31116
31073
31032
3099s
30963
30937
30917
30906
30902
IQ
32809
32758
32710
32666
32629
32598
32575
32561
32557
20
34496
34437
34381
34330
34286
34250
34224
34207
34202
21
3617S
36109
36044
3S98s
3 5934
35892
35862
35843
3S837
22
37853
37773
37699
37631
37572
37525
37490
37468
37461
23
39521
39431
39345
39268
39201
39146
39106
39081
39073
24
41183
41080
40983
40895
40819
40757
40711
40683
40674
25
42838
42722
42612
42513
42426
42356
42304
42273
42262
26
44486
44355
44232
44120
44023
43944
4388s
43849
43837
27
46126
45980
45842
45716
45607
45518
45453
45413
45399
28
47759
47595
47441
47301
47180
47081
47007
46962
46947
29
49383
49202
49031
4887s
48740
48629
48548
48498
48481
3°
Siooo
50799
50609
50437
50287
50165
S0074
S0019
50000
31
52608
52386
52177
51986
51821
51686
51586
51525
SI 504
32
54207
53964
53733
53524
53341
S3 193
53082
53015
52992
33
55798
5S53I
SS278
55048
54848
54684
54563
54489
54464
34
57379
57087
56811
56559
56340
56161
56028
55947
55919
35
58952
58634
58332
58057
57818
57622
57477
57388
57358
36
60515
60169
59841
59541
59280
59067
58909
S8811
58779
37
62068
61693
61337
61011
60727
60495
60323
60217
60182
38
63612
63206
62820
62467
62159
61907
61720
61605
61566
39
65146
64707
64290
63908
63574
63302
63099
62974
62932
4°
66671
66197
65746
65334
64974
64679
64459
64324
64279
41
68185
. 67675
67189
66745
66356
66038
65801
65655
65606
42
69688
69140
68619
68140
67722
67379
67124
66966
66913
43
71182
70594
70034
69520
69070
68701
68426
68257
68200
44
72665
72036
7143s
70884
70401
70005
69710
69527
69466
45 c
•74137 c
■ 73465 c
(.72822 c
.72232 c
■7171S c
.71289 c
). 70972 c
■70777 c
3.70711
100
ELLIPTIC INTEGRALS
III.— ELLIPTIC INTEGRALS OF THE SECOND KIND
0"
5°
10° ■
15°
20°
7
25°
30°
35°
40°
45°
46°
0.80285
0.80228
0.80056
0.79775
0.79390
0.78911
0.78350
0.77721
0.77040
0.76326
47
82030
81969
81787
81489
81081
80573
79977
79308
78584
77824
48
83776
837"
83518
83202
82770
.82231
81599
80890
80121
79313
49
85521
85453
85249
84914
84457
83887
83217
82466
81651
80794
50
87266
87194
86979
86626
86142
85539
84832
84036
83173
82265
SI
89012
88936
88709
88336
87826
87189
86442
85601
84689
83728
S2
9°757
90677
90438
90045
89507
88836
88048
87161
86197
85182
53
92502
92418
92166
91753
91187
90481
89650
88715
87698
86627
54
94248
94159
93895
93450
92865
92122
91248
90264
89193
88063
55
95993
95900
95622
95166
94S4I
93761
92843
91807
90680
89490
S6
97738
97641
97350
96872
96216
95397
94433
93345
92160
90908
57
0.99484
0.99381
0.99077
0.98576
97889
97030
96019
94878
93634
92318
S8
1. 01 229
1.01122
1.00803
1.00279
0.99560
0.98661
97602
96405
95100
93719
59
02974
02863
02529
01981
1.01229
1.00289
0.99180
97928
96560
95111
60
04720
04603
04255
03683
02897
01915
1.00756
0.99445
98013
9649s
61
06465
06343
05980
05383
04563
03538
02327
1.00957
0.99460
97871
62
08210
08084
07705
07083
06228
05158
03895
02465
1.00900
0.99238
63
09956
09824
09430
08781
07891
06776
054S9
03967
02334
1.00598
64
1 1 701
1 1 564
11154
10479
09553
08392
07020
05465
03762
01949
65
13446
13304
12878
12176
11213
10005
08577
06958
05183
03293
66
15192
15043
14601
13873
12871
11616
10132
08447
06599
04629
67
16937
16783
16324
15568
14529
13225
1 1 683
09932
08009
05957
68
18682
18523
18047
17263
16185
14832
13231
11412
09413
07279
69
20428
20262
19769
18957
17839
16437
14776
12888
10812
08593
70
22173
22002
21491
20650
19493
18040
16318
14360
12205
09901
71
23918
23741
23213
22343
21145
19640
17857
15828
I3S94
11202
72
25664
25481
2493s
24034
22796
21239
19394
17293
14977
12497
73
27409
27220
26656
25726
24446
22837
20928
18754
16356
13786
74
29154
28959
28377
27417
26094
24432
22459
20211
17731
15068
75
30900
30698
30097
29107
27742
26026
23989
21666
19101
16346
76
32645
32437
31818
30796
29389
27619
25516
23117
20467
17618
77
3439°
34176
33538
32486
31035
29210
27041
24566
21830
1888s
78
36136
35915
35258
34174
32680
30800
28565
26012
23189
20148
79
37881
37654
36978
35862
34325
32389
30086
27456
24544
21407
80
39626
39393
38698
37S50
35968
33976
31606
28897
25897
22661
81
41372
41132
40417
39238
37611
35563
33124
30336
27246
23912
82
431 1 7
42871
42137
40925
,39254
37148
34641
31773
28594
25159
83
44862
44610
43856
42612
40896
38733
36157
33209
29939
26404
84
46608
46349
45575
44299
42537
40317
37672
34643
31282
27646
8S
48353
48087
47294
45985
44178
41900
39186
36076
32623
28886
86
50098
49826
49013
47671
45819
43483
40699
37508
33963
30124
87
S1844
51565
50732
49357
47459
45066
42211
38939
35302
31360
88
53589
53304
52451
5 1043
49100
46648
43723
40369
36640
32596
89
55334
55042
54170
52729
50740
48230
45235
41799
37977
33830
90
1.57080
1.56781
1.55889
1-54415
1.52380
1. 498 1 1
1.46746
1.43229
1-39314
1.35064
FIVE-PLACE TABLES
101
III.— ELLIPTIC INTEGRALS OF THE SECOND KIND
♦
so"
55°
60°
65°
70°
75°
80°
85°
90°
46°
-75599
0.74881
0-74195
0.73564
0.73010
0.72554
0.72215
0.72005
0.71934
47
77050
76285
75553
74879
74287
73800
73436
73211
73135
48
78490
77676
76896
76177
75546
75025
74636
74396
74314
49
79920
79054
78225
77459
76786
76230
75815
75558
75471
SO
81338
80419
79538
78724
78007
77414
76971
76697
76604
51
82746
81772
80836
79971
79208
78578
78106
77814
77715
52
84143
83111
8212c
81202
80391
79720
79218
78907
78801
S3
85529
84438
8338S
82415
81554
80842
80307
79976
79864
S4
86904
85752
84641
83610
82698
81941
81374
81021
80902
SS
88269
87052
85879
8478S
83822
8302c
82417
82042
8191S
S6
89622
88340
87101
85949
84926
84076
83436
83039
82904
S7
90965
89614
88308
87092
8601 1
85110
84432
84010
83867
58
92297
90876
89500
88217
87075
86122
85404
84957
84805
S9
93619
92125
90677
89325
88119
87112
86352
85878
85717
60
94930
93362
91839
904 1 5
89144
88080
87276
86773
86603
61
96231
94586
92986
91488
90148
89025
88175
87643
87462
62
97521
95797
94118
92543
91132
89948
89049
88486
88295
63
0.98802
96996
95236
93581
92096
90848
89898
89303
89101
64
1.00072
98183
96339
94602
93041
91725
90273
90094
89879
6S
01333
0.99358
97427
95606
93965
92580
91523
90858
90631
66
02585
1.00522
98502
96593
94870
93412
92297
91595
9135s
67
03827
01674
0.99562
97564
95756
94222
93047
92305
92050
68
05060
02815
1.00609
98518
96622
95010
93771
92987
92718
69
06284
03945
01643
0.99456
97469
95775
94470
93642
93358
70
07500
05064
02664
1.00379
98298
96519
95144
94270
93969
71
08707
06173
03672
01286
99108
97240
95793
94870
94552
72
09907
07272
04668
02178
0.99900
97940
96417
95442
95106
73
1 1098
08362
05651
03056
1 .00674
98619
97016
95987
95630
74
12283
09442
06624
03919
01431
99278
97590
96503
96126
75
13460
10513
07586
04769
02172
0.99916
98141
96992
96593
76
14631
11577
08537
05607
02896
I 00534
98667
97453
97030
77
15795
12632
09478
06432
03605
01133
99170
97887
97437
78
16954
13680
10410
07245
04300
01714
0.99650
98293
9781S
79
18107
14721
11333
08047
04981
02277
1.00107
98671
98163
80
1925s
15755
12249
08839
05648
02823
00543
99023
98481
81
20399
16784
13156
09621
06304
03354
00958
99348
98769
82
21538
17807
14057
10395
06948
03870
01 3 54
99646
99027
83
22673
18825
14952
11161
07582
04372
01731
0.99920
99255
84
23805
19839
15841
11920
08207
04863
02091
I. 00168
99452
85
24934
20850
16726
12673
08825
05343
02436
00394
99619
86
26061
21857
17606
13421
09435
05813
02768
00598
99256
87
27186
22862
18484
1416s
10041
06277
03089
00784
99863
88
28310
23865
19359
14906
10642
0673s
03401
00954
99939
89
29432
24867
20233
15645
11241
07188
03708
01113
0.99985
90
I 30554
1.25868
I .21106
1.16383
1. 11838
I .07641 1.04011
1,01266
I. 00000
INDEX
Abel, Niels, 6, 7, 24, 29, 31
Addition-Theorem, 7
d'Alembert, Jean, 6
Amplitude, 24
Andoyer, H., 6
Appell and Lacour, Fonctions Ellip-
tiques, 7, 14, 21
Arithmetico-Geometrical Mean, 78
Bernoulli, Jacob, 6, 7
Burnside, W. S., 54
Byerly, Integral Calculus, 48, 63, 90, 91
Calculation of elliptic functions, 33
Calculation of elliptic function, when
modulus and argument are given, 87
Calculation of argument, when k and
the function are given, 87
Cayley, Arthur, 6, 14, 16, 28, 65, 68, 73,
76
Clifford, W. K., 41
Co-amplitude, 35
Computation, see Numerical computa-
tion
Cosine-amplitude = CH, 24
Delta-amplitude = dH, 24
Durege, Elliplische Funklionen, 59
E{k, ), see Integral E{k, 4,)
El, the complete integral, 16, 17, 81
Tables for £1, 93
Elastic curves, 6, 89
Eimepei, EUiptische Funklionen, 8, 10,
12
Euler, Leonhard, 7
Fik, h), see Integral F{k, h)
F,, the complete integral, 16, 17, 81
Tables for Fi, 93
Fagnano, G. C, 6, 7
Fricke, Elliplische Funktionen, 8
Functions, elliptic, 22-39
Gauss, C. F., 78
Graphs of the integrals of the first and
second kinds, 16, 17, 18, 19, 20
Graphs of the functions sn, en and dn,
36,37,38
Greenhill, A. G., 6, 52
Gudermann, M., 24, 28
Gudermannian, The, 28, 29
Hancock, Harris, 6, lo, 15
Hermite, Chas., 6
Humbert, G., 92
Hyperbola, rectification of, 6
Infinities, 35, 36, 37, 38
Integrals, the F{k, ) and E{k, 0), their
graphs, 15-20
Integrals, tables for F{k, >) and £(*, ) ,
94-101
Integral, the n (h, k, ip), 15
Integrals, the general elliptic, 9, 14
Integrals of certain elliptic functions,
58-63
Integrals of first kind reduced to normal
form, 41-63
Integrals, normal, 10, 15
Integrals, of the first, second and third
kinds, 9, 60
103
104
INDEX
Jacobi, C. G. J., s, 6, 7, 24, 25, 26,
29. 31, 34, 6s
Jacobi's imaginary transformation, 25,
34
K, K', tables for, 93
K, K', the transcendents, 26
Lacour, E., see Paul Appell
Lagrange, Joseph L., 7, 68
Landen, John, 68
Landen's transformations, 65—73
Latitude, 89
Legendre, Adrien Marie, 5, 6, 7, g, 10,
II, 12, 15, 21, 22, 23, 24, 61, 68, 71,
81, 82, 84,92
Legendre's transformation, 10-13
Lemniscate, 7, 88
Levy, Fonclions ellipiiques, 21, 92
Maclaurin, Colin, 6
Modulus, 12
Molk, Jules, see Jules Tannery
Numerical computation of integrals of
ist and 2d kinds, 69-87
Pendulum, simple, 90, 91
Periods, 27, 31, 35-38
Periodic functions, property of, 7, 31
Period-parallelogram, 32
Period-parallelogram of sn u, p.35,
of en u, p. 37, of in u, p. 38.
Period-strip, 31
Quadrant of lemniscate, 88; of ellipse,
89
Rectification of ellipse, 88; of lemnis-
cate, 88
Reduction formulas, 59-61
Richelot, F. S., 10
Serret, J. A., 7
Simpson's Fluxions, 91
Sine-amplitude = iK, 24
Tables of complete integrals of ist
and 2d kinds, 93
Tables of elliptic integrals of ist kind,
94-97
Tables of elliptic integrals of 2d kind,
98-101
Tables of useful integrals, 61-63
Tannery and Molk, Fonctions Eilip-
tiques, 14
Volume common to two cylinders, 89
Wallis's formula, 26, 60
Weierstrass, Karl, 24, 28
Zeroes, 35-38
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