252 Proceedings of Royal Society of Edinburgh. [sess. 
The paper closes with a reference to the case where cos X — cos //. 
= cos v — 0, and to the case where a=b—c— 0 ; the equation for 
the determination of L, M, N being in the former case 
x 3 - ( A + B + C)x 2 + (AB + BC 4- CA - a 2 - b 2 - c 2 )x 
— ABC + A a 2 + B6 2 + Cc 2 — 2 abc = 0 , 
and in the latter case 
A 2 x 3 - (A sin 2 A + B sin 2 //, + C sin 2 v)a: 2 
+ (AB + BC +■ CA)aj - ABC = 0, 
“ welche beide Gleichungen schon sonst gegeben sind.” 
Jacobi (1827). 
[De singulari quadam duplicis integralis transformatione. 
Crellds Journ ., ii. pp. 234-242.] 
Although the title of this paper is quite unlike that of the pre- 
ceding, it will be seen that the two are in essence most closely 
related. 
The double integral referred to is 
where 
// 
sin if/-dif/-dcf> 
p = a + a cos 2 t f/ + a" sin 2 i f/ cos 2 <fj + a" sin 2 ^ sin 2 <^> 
+ 2 b' cos \ ft + 2 b" sin if/ cos <£ + 2b'" sin if/ sin <f> 
+ 2c sin 2 if/ cos sin cf> + 2c ,/ cos^sin^sin</> + 2c" cos if/ sin if/ cos cf> , 
— that is to say, where p is a quadratic function of cos if/, 
sin if/ cos <j > , sin if/ sin </> ; and the purpose of the paper is to show 
that the integral can be transformed into 
ll 
sin P-0P-00 
G + G' cos 2 P 4- G" sin 2 P cos 2 # + G ' sin 2 P sin 2 # ’ 
where the denominator is a quadratic function of cos P , sin P cos # , 
sin P sin # , but contains only the squares of these quantities. The 
transformation is avowedly suggested by Gauss’ solution of a 
simpler problem of the same kind, viz., the transformation of 
f 0E 
J VL(A - cos E) 2 + (B - b sin E) 2 + C 2 ] 
into the form 
