250 
T 596 
596. 
NOTE ON A THEOREM OF JACOBI’S FOR THE TRANSFORM 
ATION OF A DOUBLE INTEGRAL. 
[From the Messenger of Mathematics, vol. iv. (1875), pp. 92—94] 
Jacobi, in the Memoir “ De Transformatione Integralis Duplicis...” &c., Crelle, t. viii. 
(1832) pp. 253—279 and 321—357, [Ges. Werke, t. ill., pp. 91—158], after establishing 
a theorem which includes the addition-theorem of elliptic functions, viz. this last is “ the 
differential equation 
drj dO 
f (G' 2 cos 2 t) + G" 2 sin 2 r\ — G 2 ) V( f? /2 cos 2 @ + @" 2 sin 2 6 — G 2 ) ’ 
has for its complete integral 
G + G' cos rj cos 6 + G" sin g sin 6 = 0,” 
[observe, as to the integral being complete, that the differential equation contains only 
the constant G 2 — G' 2 h- (G 2 — G" 2 ), whereas the integral equation contains the two con 
stants G' -r- G and G" -5- G), obtains a corresponding theorem for double integrals; viz. 
this, in the corresponding special case, is as follows: If the variables (</>, yjr) and 
(g, 9) are connected by the two equations 
= 0, 
+ a! cos </> . cos r) 
+ a" sin cf> cos y\r . sin rj cos 6 
+ a"' sin <£ sin yjr. sin 7) sin 9 
and if putting for shortness 
/3 
+ /3' COS </> . COS 7) 
+ /3" sin (j) cos . sin 7] cos 9 
+ /3"' sin <£ sin yfr . sin 7] sin 9 
0, 
a" /3'" - a"'/3" = /, a/3' - a' /3 = a, 
a'"/3' — a' /3'" = g, a/3" -a"/3 = 6, 
a' /3" — a" /3' =h, a/3'"-a'"/3 = c, 
(whence af+bg + ch = 0);