1904-5.] 
Dr Muir on General Determinants. 
909 
Five theorems with attendant corollaries are carefully formulated 
and proved, extreme simplicity and fulness of exposition being in 
evidence throughout. The first three theorems are mere variants 
of the first case of Binet’s multiplication-theorem for two non- 
quadrate matrices, viz., in later notation — 
I a i^i d* a '2 X 2 d" a 3^3 + • • • ^ll/l d~ a 2^2 d" + • • • 
I fti^i + ft^ 2 + fts X S + ’ ' ' ftlVl d" /^2 d- ft.$Z + ' ‘ ’ 
= 
°1 
a 2 
a 3 • • • • 1 
x 1 
x 2 x 3 .... 
ft 
ft 
Pz • • • • 1 
Vi 
y 2 2/3 • • • • 
a i 
a 2 
X 1 X 2 
+ | a > 
CO 
e 
x } x 3 
+ • 
...+ h 
a 3 
Pi 
ft 
\Vl V2 
Ift 
fts 
V\ y% 
I ft 2 
ftz 
The real interest arises when the a’s and /3’s of this are so 
taken that the first determinant of every pair on the right is of 
the same form as the determinant on the left, and can therefore be 
expanded in exactly the same way as the latter. The outcome is 
“ 4 e Theoreme. Soient 
P P P 
n fonctions homogenes et lineaires de n variables 
a , y , ^ , • • • • 
[viz., P x = xP xx + yP xy + zP xz + • • •, P y = • • • Jetnommons 
P P P 
ce que deviennent les fonctions P x , P y , P, , .... quand on 
remplace les n variables x , y ,z , ... par n autres variables 
x , y , z , .... 
[viz., P x = xP a , >a . + yP a . i y + zP l . fZ + * • • , P y =- • • ]. Concevons 
d’ailleurs, que l’on ajoute entre eux les termes de la suite 
p p p 
x a; 5 . 2/5 x z 5 • • • • 5 
ou de la suite 
P P P 
1 X ) f 1 Z ) 
respectivement multiplies par les variables 
x,y,z, 
ou par les variables 
x , y , z , . . . .; 
