210 Proceedings of Eoyal Society of Edinburgh. [sess. 
Four theorems he considers fundamental, viz., those known to us 
as (1) Bezout’s recurrent law of formation, in all its generality ; 
(2) Vandermonde’s proposition that permutation of bases leads to 
the same result as permutation of exponents ; (3) Laplace’s expan- 
sion-theorem ; (4) Vandermonde’s proposition regarding the effect of 
making two bases or two exponents equal. The two most important, 
viz. (1) and (2), he leaves without proof, and the 4 th he says he 
would at once deduce from the 3 rd , — doubtless by choosing the ex- 
pansion in which the first factor of every term would be of the form 
(aa , a/3) 
and therefore equal to zero. 
The proof of the 2 nd theorem, viz., 
(abc . . . r , a/3y . . . p) = ( abc . . . r , a/3y . . . p ) , 
is by the method of so-called induction, and may be illustrated in 
a later notation by considering the case 
a l 
a 2 
a 3 
oq 
h 
Ci 
= 
a 2 
h 
C 2 
C 1 
C 2 
C 3 
a 3 
C 3 
From theorem (1) we have 
c 2 c z 
a* (to 
a. 
a Y 
a 2 
a z 
h 
h 
h 
= 
a, 
c i 
C 2 
C 3 
= -\ 
— Ci 
— a 2 
h 
+ 
Cts 
\ 
c x 
C 3 
C l 
C 2 
% 
a i 
a 2 
+ b 2 
c i 
% 
b 3 
C 1 
C 2 
% 
a i 
a 2 
- c 9 
4- 
Co 
A 
\ 
V 
6 
\ 
b 2 
But by hypothesis all the determinants on the right here may have 
their rows changed into columns ; and this being done we have by 
addition and the use of theorem (1) — 
£?1 
a 2 
a 3 
a i 
c i 
\ 
h 
= 3 
a 2 
C 2 
C 2 
C 3 
a 3 
h 
C 3 
and thence the identity required. 
(IX. i) 
