1887.] Dr T. Muir on the Theory of Determinants. 
455 
Aa + A' a' + A" a" =0 
Ab+A'b' + A"b" =0 
Ac + A’c’ + A"c" + Ba + B'o' + B "a" = 0 
+ B5+B'6' + B"5" =0 
+ Bc+ B'e'+BV =0 
+ B d + B 'd' + B"ci" + C a + C 'a + C "a" = 0 
+ C6 + C'6' + C"b" =0 
+ Cc + C V + C"c" = 0 
+ C d + C 'cl' + C"d" + Da + D'a' + D "a" = 0 
+ D6 + D'&' + D"i" = 0 
+ Dc + DV + D"c" = 0 
Ad + A'd' + A "d" + Da + D 'a + D"a" = 0 
are given, and what is required is the result of the elimination 
( equation de condition ) of the twelve quantities — -a,a',a",b,b',b‘',c,c', 
c",d,d',d". This is found to he — 
(ab'c").[(bc'd") s - ( ab'c"Y-{ab'd ")} = 0. 
The two paragraphs quoted (§§ 264, 265) show that Bezout 
could obtain with considerably increased ease and certitude any one 
of Laplace’s expansions of numerator and denominator. What it 
accomplished in the illustrative example is virtually, in modern 
symbolism, the reduction of 
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