Dr Muir on General Determinants. 
679 
1907-8.] 
sketch (pp. 285-292) of the theory of determinants. Short and simple as 
this is, it contains one paragraph (§11) worthy of note, namely, in regard 
to the multiplication-theorem. 
The determinant 
Ai«i + B^£q + CjCj A 2 <q + Bf> x "t" '^'3^1 "t Bg&^ + CgCj 
Aj£/ 2 "t" 4" CjC 0 A. 2 & 2 “I” B 2 & 2 ^2^2 Ag$ 2 "t" Bg& 2 0gij?2 
Aj& 3 4 - B^?> 3 “l - Op* 3 A 2 < 2 3 4 * B 9 6 3 + 0 2 c 3 Agftg 4* Bg&g + C 3 c 3 , 
he says, is evidently the result of eliminating x, y, z from the equations 
cqSj + & 1 S 2 "t ^ 1^3 ^ 9 
a 2 Si + 5 2 S 2 4- c 2 S 3 = 0 - 
ttgSi 4- 6 3 S 2 + c 3 S 3 H 0 
when 
S T = AjJC 4- k 2 y + A 3 2 
5 2 = B 1 x + ~B 2 y + B b z , 1 
5 3 = C l x+C 2 y+Ctfs J. 
But this elimination may be effected at once by eliminating Sj , S 2 , S 3 : 
consequently | a x b 2 c 3 | must be a factor of the resultant. In the second 
place, since a set of values of x, y , 0 can be found to satisfy simultaneously 
the given equations if a set can be found to satisfy simultaneously the 
equations S x = 0, S 2 = 0, S 3 = 0 : and since the condition that the latter shall 
be possible is j A 1 B 2 C 3 i = 0, it follows that i A 1 B 2 C 3 | must also be a 
factor of the result. The remaining factor being manifestly 1, the desired 
end, in Salmon’s opinion, is attained. We only remark in passing that a 
little careful scrutiny of the reasoning would have suggested the need for 
additional support. 
Salmon also proposes a fresh enunciation of the same theorem, namely. 
If any set of linear equations 
a x x 4- byj + c x z 4- .... = 0 
a. 2 x + b 2 y + c 2 z + . . . . = 0 - 
be transformed by any linear substitution 
x = Aj^ + B-^ + cy + . . . . "j 
y = A 2 £+ B 2 ?7 4 C 2 £-f 
then the determinant of the new set will be equal to the determinant of the 
original set multiplied by the determinant of transformation. This new 
wording will be recognised as a sign of the advent of the “ algebra of linear 
transformation.” 
