474 Proceedings of Eoyal Society of Edinburgh. 
[JULY 18 , 
aQ + /?S — yP = 0 ] 
SP +aR = 0 I 
-yN +SS + aM = 0 j 
— /3M + yft + 8Q = 0 J , 
or 
— \b- l c 2 d^ [ .\a 1 d 2 e^ + {a^dflbft^e^ — \afb 2 d^ [ .\c l d 2 e ^ 
\ a \ ^2 C 3l • l c i^2 e 3l l^l C 2^sl • \ a i C 2 G ^[ ~ i^l C 2^3l • l^i c 2^3l 
— . 1 ^ 1 ^ 363 ! + I^VsI-IVW “f" l^l C 2^3l'l tt 1^2 e 3i 
— I^Cg^gl. + | tt i^2^3l- l a i C 2 e 3l “ \ a i^2 e ^ 
= 0 
= 0 
= 0 
= 0 
1 
[- (xxm. 2 .) 
1 
J 
which in their turn, he says, by processes of elimination, may he 
the source of many others. For example, each of the four being 
linear and homogeneous in a, /3, y, 8 , these letters may all he 
eliminated with the result 
PuS + QISr-PM = 0, 
or 
\ a i C #A • l^l^2 e 3l “ l^l C 2%l — l^l^3! • 1^1 ^2^1 “ 
Also, eliminating P from the first and second, S from the first and 
third, Q from the first and fourth, and so on, we have 
- j3yN + 8aQ + /3SS + ay ft = 0 , 
a/3M + y8P — /3yN — 8aQ = 0 , 
a/5M — ySP + /38S — ay It = 0 , 
&c. &c. 
i.e. 
J ct]Cft% |-| ei-J^d's |*1 b iC 2 e 3 ] 
1 tt 1^2 C 3 1 • 1 
VA 1 ’l 
jH- 
| a^cfL^ j.j af> 2 C 3 |.| bfl$ % 1 "f 
| b 1 c 2 d 2> |.| 
“iMsI-l 
&c. 
&c. 
(xxvm.) 
Monge does not pursue the subject further. His method, how- 
ever, is seen to he quite general ; and we can readily believe that 
he possessed numerous other identities of the same kind. This is 
borne out by a statement in Binet’s important memoir of 1812. 
Binet, who was familiar with what had been done by Vandermonde, 
Laplace, and Gauss, says (p. 286) : — “M. Monge m’a communique, 
depuis la lecture de ce memoir©, d’autres theoremes tres-remarquables 
sus ces r 6 sultantes ; mais ils ne sont pas du genre de ceux que nous 
nous proposons de donner ici.” 
