833] ON A FORMULA IN ELLIPTIC FUNCTIONS. 293 
which are themselves at once deducible from formulae given, p. 63, of my Elliptic 
Functions, and which may be written 
sn (u, + u 2 ) = si 2 — s.f = — (c 2 — ci) = — ^ (cl 2 — d£), 4- (s 2 c 2 d 2 — s.p-pl,), 
cn (u 4 + ii 2 ) = s^cL — s. 2 c 2 d,, 4- „ 
dn (u 4 + U 2 ) = SjdjCa — S 2 d 2 c 1 , 4- „ 
In fact, the numerators of cn (u, + u. 2 ) — dn (u 4 + u 2 ), cn (u, + u 2 ) + 1, dn (u Y + u 2 ) + 1 
thus become = (s 4 + s 2 ) (c 4 d 2 — c 2 d l ), — + c 2 ) (djS* — cLs,), (d, + d. 2 ) (s x c 2 — s 2 Ci) respectively: 
so that, taking the numerator of sn (u 4 + u 2 ) successively under its three forms, we 
have by division the formulae in question. And then, if u 4 + u 2 = — (u 3 + u 4 ), the functions 
on the left-hand side become, with only a change of sign, the like functions of ii 3 + u 4 ; 
and we thence have the required equations 
•§1 ^2 *3 S4 
c,cL — c.d, c 3 d 4 — c 4 d 3 ’