288 Proceedings of Royal Society of Edinburgh. [sess. 
Jacobi (1832). 
[De transformatione et determinatione integralium duplicium 
commentatio tertia. — Credo's Journ ., x. pp. 101-128.] 
This memoir, although classed by its author with the two others 
of which we have given an account, is of much less interest on the 
purely algebraical side. In fact it consists almost entirely of the 
transformation of integrals like 
VR sin <£ d<t> cty , [[ d<t>a<lr, 
J J vR 
by means of substitutions like 
m cos cf> 
cos,?= ^/fr 
sin y cos 0 = 
n sin <t> cos if/ 
Vr 
sin 7] sin 0 = 
p sin § sin if/ 
Jr 
where 
R = m 2 cos 2 cf> + n 2 sin 2 </> cos 2 if/ + p 2 sin 2 cf> sin 2 if / . 
When, however, an advance is made from R to U, i.e. to 
a 2 cos 2 <£ + b 2 sin 2 <j> cos 2 i {/ + c 2 sin 2 cf> sin 2 if/ + 2d sin 2 </> cosi f/ sin if/ 
+ 2e cos <f> sin c£ sin ^ 4- 2 f cos <f> sin </> cos i f/ , 
the underlying algebraical problem becomes of more importance ; 
for example, such a problem (p. 122) as the finding of the co- 
efficients of the substitution 
u = gx + hy + iz A 
v = g'x + h'y + tz , t 
io = g'x + h"y + x'z , J 
which transforms 
ax 2 + by 2 + cz 2 + 2 dyz + 2 ezx + 2 fxy , 
ax 2 + b'y 2 + cz 2 + 2 d'ryz + 2 ezx + 2 fxy, 
into 
u 2 + v 2 + w 2 , 
u 2 v 2 io 2 
m\ n 2 p l ’ 
respectively. Still there is nothing calling for more than this 
passing mention. 
(. Issued separately A ugust 18 , 1902 .) 
