C. XI. 
9 
[723 
723] 
VARIOUS NOTES. 
65 
° 2 ) 2 — o, 
other integral 
On a Formula in Elliptic Functions : p. 127. 
Cil 'll 
Writing en u = , then the formulae p. 63 of my Elliptic Functions give 
rp rpr ß ß/ 
sn (u + v) = Q _ -ß,, en (u + V) = G _ Q, ; 
and, substituting for T, T', B, B', and G, C' their values, we obtain 
. s sn u en v 4- sn v en u 
sn (u + v)= - -J- , 
1 + k- sn u en u sn v en v 
en (u + v) = 
en u en v — sn u sn v 
1 — /c 2 sn u en u sn v en v ’ 
formulae which, as regards their numerators, correspond precisely with the formulae, 
and 
sin (u + v) = sin u cos v + sin v cos u 
cos {u + v) = cos u cos v — sin u sin V, 
of the circular functions, and which in fact reduce themselves to these on putting k = 0. 
The foregoing formulae, putting therein lc~ = — 1, are the formulae given by Gauss, 
Werke, t. ill., p. 404, for the lemniscate functions sin lemn (a + b) and cos lemn {a + b) ; 
where it is to be observed that these notations do not represent a sine and a cosine, 
but they are related as the sn and en, viz. that 
cos lemn a = V(1 — sin lemn 2 a) 4- V(1 + sin lemn 2 a).