1899 - 1900 .] Dr Muir on the Theory of Alternants . 
109 
when there only remains to continue the selfsame process upon the 
alternant of lower order now reached. 
It may be remarked in passing that the identity 
\a"Pc 2 d 3 \ = d8| a 0£L c 2| _ + d\a°b 2 e 3 \ - |aW| , 
which expresses the alternant in descending powers of d , when 
taken along with the identity known to Cauchy 
[a% W| = (d _ c )(d - b)(d - a)(c - b)(c - a)(b - a) 
the right side of which may likewise be arranged in descending 
powers of d, viz., 
{d 3 - d 2 (a + b + c) + d(ab + ac + be) - abc}(c - b)(c - a)(b - a ) , 
may have been the means of suggesting to Schweins his theorem 
regarding alternants like \a% 2 c 3 \ , JaWc 3 ! which have one break 
in their series of indices. In other words, the order in which he 
gives his theorems was very probably not the order of discovery. 
The remaining portion of the chapter is an investigation of the 
quotient of two alternants of infinite order, viz., 
II . a+h a+2h 
1 B A x A 2 . . 
. a+(n - 1 )h . a+nh 
• * A n+1 ' ' 
°°\ 
. . A ) 
oo 
I] . a . a+h . a+2 h 
II ^1^2 “^3 ' ‘ 
0° N 
. . A ) 
00 x 
SYLVESTER (1839). 
[On derivation of coexistence : Part 1 , Being the theory of simul- 
taneous simple homogeneous equations. Philos. Mag., xvi. 
pp. 37-43.] 
As has been already shown, Sylvester’s first approach to the 
subject of determinants was similar to Cauchy’s, the bases of both 
being the outward resemblance of the two expressions 
be 2 + a 2 e 4- ab 2 - a 2 b — ac 2 - b 2 c , 
b x c 2 + a 2 e x + a x b 2 - a 2 b x - a x c 2 - b 2 c x . 
As the former is equal to 
(c - b)(c - a)(b - a) or PD(a6c), 
