374 EXPRESSIONS FOR PL ANA’S e, y IN TERMS OF THE ELLIPTIC e, y. [467 
whence, adding, we have the first equation. 
And, moreover, 
7 0- + f?s m3 ~ 5i2 mi ) = 7 ( 1 ~ tvs mS + sh m * 
+ t¥s m3 ~sh mi ) 
+ 7^ 2 ( IE m% ) 
+ y*e (- A) 
+ 7 3 ( ~ th m ~) 
+ 7 3 e 2 (A) 
+ 7 5 (I) 
+ 7e' 2 ( 
- 
9 
S' 
m 2 ), 
rye 2 
(- 
1 - 
W O 
= 7 e 2 (- 
1 + 
3 
¥ 
m 2 
-• 
1111 
2 56 
m 2 ) 
+ 7 3 e 2 ( 
- 
ye 4. 
(tf) 
= ye* (tg 
■). 
7 3 
(- 
3 4- 
8 W 
■TVS O 
= 7 3 (“ 
| + 
5 
128 
m 2 ) 
7 3 e 2 ( 
23"! 
3 2/ 
= 7 3 e 2 ( 
23\ 
3 2/ 
7 5 
( 
if) 
= 7 5 ( 
1£\ 
04/ 
ye' 2 
( 
f m 2 ) 
= 7e' 2 ( 
9 
s 
w 2 ), 
whence, adding, we have the second equation. 
It may be noticed that, taking the foregoing expressions only as far as the third 
order, we have 
Plana. Elliptic. 
e = e (1 + i 7 2 - f m 2 ), 
7 = 7- 
And moreover that, attending only to the terms which are independent of m, 
we have 
e = e (1+lf - | f +1 7 4 ), 
7 = 7 i 1 “ Wi e< + A e2 7 2 “l7 4 )- 
which are formulae that may be found useful.