562 Proceedings of Royal Society of Edinburgh. [sess. 
Jacobi (1831 Dec.). 
[De transformatione integralis duplicis indefiniti 
00 00 
A + B cos 0 + C sin 0 + (A' -f B' cos 0 + C' sin 0) cos 0 + ( A" + B" cos 0 + C" sin 0)sin 0 
in formam simpliciorem f — — 7 ^ ^ j . 
J G - G cos y] cos 6 - G sin g sin 6 
GrelMs Journ ., viii. pp. 253-279, 321-357.] 
As the algebraical transformation effected in this paper is an 
extension of that dealt with in Jacobi’s second paper of 1827, it is 
only what might have been expected to find expressions contained 
in it which may be viewed as axisymmetric determinants. Such 
expressions are two forms of the square of 
A'(B'0'j - B"C') + B(C'A" - C"A') + C(A'B" - A"B') , or A, 
and the non-zero side of the cubic equation therewith connected, 
upon which the whole investigation depends, viz., 
z 3 - x 2 {A 2 + B 2 + C 2 + A' 2 + B' 2 + C 2 + A" 2 + B" 2 + C" 2 } 
+ x {(B'C" - B"C') 2 + • • • • } 
- {A(B'C " - B"C') + B(C'A" - C"A') + C(A'B" - A"B')} 2 . 
No hint, however, is given of these expressions being deter- 
minants, — a fact which is all the more noteworthy in view of the 
reference made in the second paper of 1827 to the writings of 
Laplace, Vandermonde, . . . , and in view of the reference made 
on p. 350 of his present paper to Cauchy’s of 1829, where, as we 
have just seen, “ fonctions alternees ” are explicitly used throughout. 
As a mere aid to the memory it would appear to have been worth 
while to note that if one of the said squares of A be the deter- 
minant formed from 
l n m 
n m V 
m V n 
the non-zero side of the fundamental cubic is the determinant- 
formed from 
x — l n m 
n x — m V 
m T x — 7i' i 
