[658 
658] ON SOME FORMULAE IN ELLIPTIC INTEGRALS. 147 
8 
to transform 
by that letter : 
k of ambiguity). 
in virtue of the foregoing values 
7i-«i = ^(.£ — 8)(ry- a) and 7l - = 
Moreover 
/,2__a-fi-y-8 _ «i- 
ry ~ a. ¡3 — 8 7i-«i- $i— 8 X ' 
Hence the like formulae with the same value of k 2 , and with 2a in place of a, will 
be applicable to the like differential expression in x x : viz. assuming 
a x sn 2 u x — 8 X sn 2 2a 
sn 2 u x — sn 2 2a 
we have 
dx x 
- 2 
\/ x x — a x . x x — (3 X . x x — y x . x x — Sj V 7l — a x . /3 X — 8 X 
du x . 
We have thus the integral of the differential equation 
dx x 
\/x x — a x . x x — fi x . x x — 7l . x x — 8 X 
dx 
x — a. x — ¡3.x — <y .x — 8 
(the two quartic functions being of course connected as before); viz. assuming x, x x 
functions of u, respectively as above and recollecting that y 2 — a x . ¡3 X — 8 X — y — a. (3 — S, 
we have du x = du; and therefore u x = u + f ( f an arbitrary constant); the required 
integral is thus given by the equations 
sn 2 u _ x — 8 _ 
sn 2 a x — a ’ 
sn 2 (u+f)_x x — 8 X ' 
sn 2 2 a x x — a 1 i 
( f the constant of integration). 
Using the formula 
sn (w+/) = 
sn 2 u — sn 2 / 
we obtain 
x x 8 X 
sn 2 2 a : 
sn u cn /dn /— sn /cn u dn u ’ 
{(x — 8) sn 2 a — (x — a.) sn 2 f } 2 
x x — a x {V# — a . x — 8 sn a cn/dn f—*Jx — ¡3.x —<ysn /cnadna} 2 ’ 
which is the general integral. 
We obtain a particular integral of a very simple form by assuming /= a, viz. 
this is 
^■sn=2a- Snaa 
(a - 8f 
X-I — ct 
this is 
cn 2 a dn 2 a (V# _ a. x - 8 - \/x - ¡3. x - 7 } 2 ’ 
x x — Si 7i — «1 7 — ct. /3 — 8 
x x — ct x 7i — x — % .x— 8 — \/x — (3. x — ry} 2 ’ 
19—2