564 Proceedings of Royal Society of Edinburgh. [sess. 
Jacobi (1833). 
[De binis quibuslibet functionibus homogeneis secundi ordinis per 
substitutiones lineares in alias binas transformandis, quae solis 
quadratis variabiliiun constant ; una cum . . . Crelle’s 
Journ ., xii. pp. 1-69.] 
As in this great memoir Jacobi sums up and generalises the 
results of his papers of 1827, 1831, 1832, in which, as we have 
seen, axisymmetric determinants were implicitly made use of, it 
is at first somewhat surprising to find very little reference to 
properties of determinants of this special form. The reason, 
however, doubtless is that when he came to extend his theorems 
from the third or fourth order to the wth, he also withdrew the 
restriction as to axisymmetry and gave the results in quite general 
form. In support of this the fifth and sixth sections (pp. 8-11) 
may be referred to, — sections which on account of being concerned 
with determinants in general have already been dealt with in the 
proper place. Even when he comes, as before in the particular 
cases, to his equation for determining the coefficients of the 
squares of the new variables, that is, the equation 
r = o 
where V is described as the expression got from 2 ± a u a 22 . . . a nn 
by changing a u , a 22 , . . . into a n -x, a 22 -x, . . . he gives an 
expansion of T according to ascending powers of x, which holds 
whether a K ^ = a\ K or not. The passage is — 
“ Quod attinet ipsam ipsius V formationem, observo, si 
signo summatorio S amplectamur expressiones inter se 
diversas, quse permutatis indicibus 1, 2, 3, . . . , n pro- 
veniunt, fieri : 
M 
1+ 
K> 
. . a nn 
- 
x S 2 ± a n a 22 . , 
• • 0>n-l,n—l 
+ 
x 2 S1.± a u a 22 . 
• • ^n— 2, n — 2 
+ 
x n ~' 2 S 2 ± a n a 22 
+ 
x n ~ x S 2 + u 11 
+ 
x n . 
