419 
1888-89.] Dr T. Muir on the Theory of Determinants. 
The fourth example concerns the elimination of x, y, z between 
the three equations 
Ax 2 + B if + Cz 2 + 2 A 'yz + 2B 'zx + 2C 'xy = 0 
Lx 2 + My 2 + Nz 2 + 2L 'yz + TWzx + 2 Wxy = 0 
Vx 2 + Q.y 2 + R z 2 + 2P 'yz 4- 2Q 'zx + 2R 'xy = 0 
Using each of the three multipliers x> y , z with each of the three 
equations, we obtain nine equations linear in the ten quantities, 
x s , y 3 , z 3 , x 2 y, x 2 z, y 2 x, y 2 z, z 2 x, z 2 y , xyz . 
Another such equation is thus necessary for success. Sylvester 
obtains it very ingeniously by writing . the given equations in the 
form 
(A# + B 'z +C'y)x + (By + C'x +A!z)y + (Cz h-A'^ + B'^ =0^ 
(Dr + M'^ + ^'y)x + (July + Wx + Uz)y + (Nz + L'y + Wx)z = 0 > 
(Px + Q 'z + R 'y)x + (Q y + R'^r + P 'z)y + (Rz + P 'y + Q 'x)z = o) i 
and then eliminating x, y , z. The work is not continued further. 
We may ourselves note, in conclusion, that the fourth example 
includes in a sense the three others, but that it does not follow 
therefrom that by giving the requisite special values to the co- 
efficients in the result of the general example, we should obtain the 
results for the particular examples in the forms already reached. 
Indeed, it is on account of this apparent non-agreement that the 
dialytic method is valuable to the theory of determinants, some very 
remarkable identities being arrived at by its aid. An explanation 
is also thus afforded of the trouble we have taken to elucidate its 
history. 
CRAUPURD, A. Q. G* (February 1841). 
[On a method of algebraic elimination. Cambridge Math. Journal , 
ii. pp. 276-278.] 
In Craufurd we have an independent discoverer of the dialytic 
method. A full account of his paper is quite unnecessary : the few 
* Only the initials A. Q. 0. C. are appended to the article. There can be 
little doubt, however, that they belong to Craufurd, whose name in full 
appears elsewhere in the Journal. 
