1887.] Dr T. Muir on the Theory of Determinants. 
473 
/ transmutatum iri per substitutionem 
aS + f3&' + yS", ae + /3t + ye' at, + ftf + y£" 
a 8 + /TS' + y'8" a'e + 0V + y'e" af + /3'£ + yf" 
a '8 + p"8' + y"8" ae + /TV + y V ' a"£ + + y (xxn. 3. ) 
MORGE (1809). 
[Essai duplication de l’analyse a quelques parties de la geometrie 
elementaire. Journ. de VEc. Polyt., viii. pp. 107-109.] 
Lagrange, as we have already seen, was led to certain identities 
regarding the expression 
xy'z" + yzx' + zx'y" — xtfy" - yx f z" — zyx" 
in the course of investigations on the subject of triangular pyramids. 
The position of Monge is that of Lagrange reversed. Erom the 
theory of equations he derives identities connecting such expressions, 
and translates them into geometrical theorems. 
The simpler of these identities, as being already chronicled, we 
pass over. At p. 107 he takes the three equations 
ayx 4 - b x x + cyy + d^z + e 1 = 0 
a 2 u + b 2 x + e 2 y + d 2 z + e 2 — 0 
a 3 u + b 3 x + c 3 y + d 3 z + e 3 = 0 , 
and eliminating every pair of the letters u, x, y, z, obtains the six 
equations 
pu 4- ax + P = 0 
(1) 
yx + Py + Q = 0 
(2) 
8y + yz + M = 0 
( 3 ) 
az + 8u + N = 0 
( 4 ) 
yu - ay + S = 0 
(5) 
Pz - Sx + R = 0 
( 6 ); 
the ten letters 
a, /?, y, 8, M, E", P, Q, R, S 
being used to stand for the lengthy expressions which we nowa- 
days denote by 
I b-^e 2 d 3 | , [ ayz^d 3 j , | afj,~)d 3 ( , | af)c)C 3 j , 
I ^1^2^3 I J i I 5 I h^2^3 I 5 — | ^1^2^3 I "> I ^1^2^3 I ’ i ^1^2% I ' 
Then, taking triads of these six equations, e.g ., the triads (1), (2), 
(5), he derives the identities 
