566 
Proceedings of the Royal Society 
function vanishes identically, and we note particularly that the 
basis of the proof is that the interchange of the two indices in 
question changes the sign of the function, and yet leaves the 
function unaltered. 
Upon this theorem the solution of a set of simultaneous linear 
equations is then with much neatness made to depend. In more 
modern notation Vandermonde’s process is as follows : — It is known 
that 
a 1 | \e 2 | + \ | c x a 2 | + c 1 \ af 2 | = | 1=0, 
and 
a 2 1 ^1 C 2 1 
+ \ | c x a 2 
| + c 2 | af> 2 
1 = 
1 a 2^] 
a 1 1 
1 1 »A 1 
+ aj Cl “ 2 
+ 
= 0 
and 
„ IV 2 i 
2 i «A 1 
. ft 1 C \ a 2 1 
2 1 1 
+ c 2 
= 0 
hence, if the equations 
a Y x + b Y y + e 1 = 0 
a 2 x 4 - b 2 y + c 2 = 0 
he given us, we know that 
x = 
h _A ", l e Al 
y 
is a solution. 
This result, moreover, is generalised ; the solution of 
r x x x + r 2 x 2 + +r n x n + r n+1 = 0 (r=l,2,...,n) 
being fully and accurately expressed in symbols, although the 
numerators of the values of aq, x 2) . . . , x n are not in so simple a 
form as Cramer’s rule for obtaining the numerator from the de- 
nominator might have suggested. 
Lastly, and almost incidentally, Vandermonde makes known a 
case of the widely general theorem now-a-days described as the 
theorem for expressing a determinant as an aggregate of products of 
complementary minors. His case is that in which the given de- 
terminant is of the order 2m, and one factor of each of the products 
is of order 2. 
Summing up, therefore, we must put the statement of our in- 
debtedness to Vandermonde as follows : — 
