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A THEOREM IN ELLIPTIC FUNCTIONS. 
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[From the Proceedings of the London Mathematical Society, vol. x. (1879), pp. 48 48. 
Read January 8, 1879.] 
The theorem is as follows: 
If u + v + r + s = 0, then 
1 k' 2 
— k 2 sn u sn v sn r sn s + cn u cn v cn r cn s — j- dn u dn v dn r dn s = 
k 2 k 2 
It is easy to see that, if a linear relation exists between the three products, then 
it must be this relation: for the relation must be satisfied on writing therein 
v = — u, s = — r, and the only linear relation connecting sn 2 u sn 2 r, cn 2 u cn 2 r, dn 2 u dn 2 r 
is the relation in question 
1 ‘ k' 2 
— k' 2 sn 2 u sn 2 r + cn 2 u cn 2 r — y- dn 2 u dn 2 r = — y—. 
k 2 k 2 
A demonstration of the theorem was recently communicated to me by Mr Glaisher; 
and this led me to the somewhat more general theorem 
— k' 2 sn (a + /3) sn (a - /3) sn (7 + 8) sn (7 — 8) 
+ cn (a. + /3) cn ( a ~ /3) cn (7 + 8) cn (7 — 8) 
— dn (a + j8) dn (a - /3) dn (7 + 8) dn (7 — 8) 
Hj 
k' 2 2k' 2 (sn 2 a — sn 2 7) (sn 2 /3 — sn 2 8) 
k 2 1 — k 2 sn 2 a sn 2 /3.1 — k 2 sn 2 7 sn 2 8'