1888 - 89 .] Dr T. Muir on the Theory of Determinants. 40 7 
the four given equations,” the process being an instance of what he 
would similarly term the “ derivation of coexistence.” 
By proper choice of the arbitrary quantities it may be readily 
shown, as Sylvester proceeds to do, that the theorem gives (1) the 
result of the elimination of n unknowns from n equations ; (2) the 
two equations of condition in the case of n + 1 equations connect- 
ing n unknowns; (3) the ratio of any two unknowns in the case of 
n— 1 equations connecting n unknowns ; and (4) the relation 
between any three unknowns in the case of n - 2 equations connect- 
ing n unknowns. For example, the equations being 
a x x + b^y + CjZ = 0 
a 2 x + b 2 y + c 2 z = 0 
a 3 x + b 3 y + c 3 z = 0 
the theorem gives the general derivative 
«1 fl Ol 
\ fi Oi 
c i fi 9i 
«2 fi 9i 
X + 
^‘2 fl Os 
y + 
G 2 f% 92 
<h /s 3s 
oF 
w 4 * 
05 
C 3 f 3 93 
which is true whatever f v f 2 , / 3 , g v g 2 , g 3 may be. By putting 
fi, fv U 9v 9v 9v = K K K c v c v c 3> this takes the form 
\a-J) 2 c^x -i- \b-J) 2 c<fy + \c-J) 2 c^z = 0 , 
whence the equation of condition, or resultant of elimination, 
|<*i Vsl = 0 . 
As a corollary to one of the deductions from the leading theorem, 
— the deduction numbered (3) above, — the following proposition of 
a different character is given (p. 42) : — 
“ If there be any number of bases ( abc . . . Z), and any 
other, two fewer in number, ( fg . . . &), 
£PD (afg . . . h) x £PD (be . . . 1) 
+ £PD (bfg . . . h) x £PD(ac . . . 1) 
+ £PT)(afg . . . Jc) x £PD (be . . . 1) 
+ £P D(lfg . . . 1c) x ^ T>(abc. . . ) = 0, 
