1902 - 3 .] Dr Muir on Axisymmetric Determinants. 
569 
Expanding [n, i~\ after the manner of D above, but now in 
terms of the elements of the last column , we obtain 
[n, i] = A n _ h 
n, i 1 
ra — 1 , raj 
n, i 
w-2 ,n 
+ A. 
I” n > i 
\_n - 3 , n 
and therefore, since the second factors on the right do not contain 
A n _i, n or A n> n _ x (both the % th row and n lh column being gone in 
all of them), there results 
d\_n, (\ 
dA„ 
Substituting this above we see that 
dA 
n,n - 1 
n,n - 2 
n - 1, n 
+ A, mi _ 2 
n — 1, n 
dD r in » 
- = - | [ra,ra - 1J - A n>n _ x 
= - \n } n - 1] - \n,n- 1] , 
= - 2 \n,n - 1] . 
The theorem having thus been proved for the case of the suffixes 
(n - 1, n), the passage to the case of any unequal suffixes is made 
by saying “ Par un deplacement de series horizontales et de series 
werficales, on trouvera 
dD 
- (-l) i+9 2 [i,g] 
d Kg 
comme il est dit dans renonce. ,J 
Save for a page in which the development of a symmetric 
determinant for the cases n = 2, 3, 4 is given, the rest of the 
paper is taken up with the concluding portion of the solution of 
the problem of transformation. It may be well to note, however, 
that on the page referred to (p. 347) the determinant of the 
system 
U2 
A 1b 
A 22 ~ U 
... 
• • » A-o n , 
- u 
is denoted by 
det. [ A u - u, A 22 - u , . 
PROC. ROY. SOC. EDIN. — YOL. XXIV. 
• , Kn ~ U] 
37 
