394 Proceedings of Royal Society of Edinburgh, [june 18 , 
from which it follows that 
a{h cd ... l)^ - h[a cd ... l)^ 
= a{h{cd . . . T)^_^-\-{cd . . . l)^} 
-b{a{cd . . . l\_-i^ + {cd . . . l)fj 
= {a-h){cd . . . l)^ (a). 
Now consider the alternant 
1 ap^ (ctPrY • • • {ci>PrY~^} 
where A' is symmetric with respect to c,d, . . . I, and may or 
may not involve a, and where is, for shortness, written for 
(bed . . . l)^. 
The complementary minor of A' is the difference-product of 
b(acd . . . Z),., c(abd . . . . ... , l(abc . . . 7c)^; and there- 
fore by (a) is equal to 
n^(a be... hY^i{b cd ... 1), 
IIj consisting of - 1) (7^ - 2) factors, being that part involving a 
of the product of the sums of the products (r at a time) of the 
letters taken ^ - 2 at a time. 
Thus the alternant 
1 a .. . ITj(a b c . . . h)^ . A' 
■ ■ (X). 
In this identity put A' = cd . . . Z)” \ The left-hand side 
becomes the difference-product of 
a(b ed . . . Z),., b(a cd . . . Z)^, . . . , l(abc ... 
which by (a) is equal to 
7r(a be... h\ ^(a be ... 1), 
7r(abc . . . li)^ being the complete product of the sums of the 
products (r at a time) of the letters taken - 2 at a time, and 
having 1) factors. 
Hence, dividing off by 'ir(abc . . . we have 
