FOKM OF TANGENTIAL EQUATION. 
383 
from equation (64) we have 
s= — fjcot* <p sec 2 <p cosec <p d<p — § Jcot 3 <p sec 2 <p cosec <p d<p. . . . (74) 
We reduce the first of these integrals to the normal form of elliptic integrals as 
follows : — 
Let 3=cot § <p, and we find 
J cot* <f sec 3 f cosec <?<?? = 
| Vl+r 3 qJ 
1 nf 
sirdpcos^p 4 J 
zdz 
zdz 
vi+^ 
F(i, «)-f E(*, 0), 
(75) 
where & and <p are connected by the equation 
cos4=° osl| P- ( Vjj-DsMp (761 
cos^ <p + ( V3 + 1) sird <p 
The second integral in equation (74) may be derived from the first by changing the 
sign and putting 0— for <p. Hence we have at once 
Jcot* <p sec 2 <f> cosec <p d<f> 
1 3& 
= ~t sin^cos^+Y C0t & 
where S' is given by the equation 
F( i, e (W), 
(77) 
Slid <p — ( \/S — 1) cos » <p 
cos I ;/ — - — — r • 
sndcp + ( \/3 + l)cos^ <p 
(78) 
and substituting from equations (75) (77) in (74), we get the required intrinsic equation 
cos ^<p — siid<p 
siid <p cos® <p 
+ 31) cot A0 . A(0) — cot $0 . AS'} 
_^^ ]f(M -f(M')} 
+ 3qE(M)-E(M’)}, J 
(79) 
where Jc= 
\/3 + 1 
2 V 2 
, and 0, 0' are given by the equations (76), (78). 
Section III . — Transformation of the Intrinsic into the Tangential Equation. 
33. We shall have much use to make of the intrinsic equation of the catenary in this 
mdccclxxvii. 3 i 
