596] 
251 
on Jacobi’s transformation of a double integral. 
-R 2 = f 2 ( sin ф COS л/г) 2 (sin ф sin л/г) 2 
+ g 2 (sin ф cos л/г) 2 (cos ф) 2 
+ h 2 (cos ф) 2 (sin ф cos л/г) 2 
— a 2 (cos ф) 2 
— b 2 (sin Ф COS л/г) 2 
— C 2 (sin Ф sin л/г) 2 , 
$ 2 = f 2 (sin 7] COS в) 2 (sin 7] sin в) 2 
+ g 2 (sin 7] sin в) 2 (cos 7j) 2 
+ h 2 (cos 7]) 2 (sin 7] cos df 
— a 2 (cos 7]) 2 
— b 2 (sin 7J cos в) 2 
— c 2 (sin 7] sin в) 2 , 
then we have 
sin ф с1ф d-ф _ sin 7) ¿7] dd 
R “ S * 
And it may be added that the integral equations are, so to speak, a complete 
integral of the differential relation; viz. in virtue of the identity af+bg+ch = 0, the 
differential relation contains really only four constants; the integral relations contain 
the six constants a : a : a" : a'" and ¡3 : ¡3' : f3" : 0", or we have two constants 
introduced by the integration. 
The best form of statement is, in the first theorem, to write x, у for cos y, sin?;, (x 2 -\-y 2 = 1), 
7] for cos 6, sin в, {0- + y 2 = 1), and similarly in the second theorem to introduce the 
variables x, y, z connected by x 2 + y 2 + z 2 = 1, and £, y, £ connected by I- 2 + y 2 + £ 2 =1 ; 
then in the first theorem dy, dd represent elements of circular arc, and in the second 
theorem sin ф dф d-ф and sin у dy dd represent elements of spherical surface, and the 
theorems are : 
I. If (x, y) are coordinates of a point on the circle x 2 + y 2 = 1, and (f, y) coordinates 
of a point on the circle £ 2 + ?? 2 =1, and if ds, da are the corresponding circular elements, 
then 
ds _ da 
*J(ax 2 + by 2 - c) ~ VO! 2 + by 2 - c) ’ 
has for its complete integral 
ах% + byy — c = 0. 
32—2