292 
[833 
833. 
ON A FORMULA IN ELLIPTIC FUNCTIONS. 
[From the Messenger of Mathematics, voi. xiv. (1885), pp. 21, 22.] 
Writing s, c, d for the sn, cn, and dn of an argument u, and so in other cases: 
we have s, c, d for the coordinates of a point on the quadriquadric curve x 2 + if = 1, 
z 2 + k 2 x 2 = 1. Applying Abel’s theorem to this curve, it appears that, if u x + u,, + u 3 + u 4 = 0, 
the corresponding points are in a plane; that is, the elliptic functions satisfy the 
relation 
Si, Ci, di, 1 j = 0. 
#2, C2, d 2 , 1 
S3, c 3 , d 3 , 1 
s 4 , c 4 , d 4 , 1 
This may be written 
($2 '?i) (c 3 d 4 c 4 d a ) A (§4 s 3 ) (Cjdo c 2 d 1 ) 
+ (c 2 Cj) (d A s 4 d 4 Sg) + (C4 c 3 ) (diS., d^s l ) 
+ (d 2 - di) (s s c 4 - S4C3) + (d 4 - d 3 ) (sa - s 2 c x ) = 0 ; 
and it may be shown that each of the three lines is, in fact, separately = 0. 
This appears from the following three formulae: 
sn (u 4 + u 2 ) 
cn (iij + u 2 ) — dn (u 2 + u 2 ) 
sn (Ui + u 2 ) 
cn (iti + u. 2 ) + 1 
sn {til 4- u 2 ) 
dn (ili + w 2 ) + 1 
S 1 S 2 
Cid 2 Cod] 
Ci c 2 
d 1 So d 2 Si 
jç 2 (di d 2 ) 
SiC 2 s 2 c 1