Dr T. Muir on the Theory of Determinants. 
497 
In Cauchy’s theorem the corresponding numbers were found to he 
and P being not any whole number as s is, but like C a 
combinatorial. Without further investigation, we might conse- 
quently assert that, supposing the two theorems to be alike in other 
respects, Binet’s must be the more general, the passage from it to 
Cauchy’s being effected by taking s = C. A closer examination, 
however, will show that this is not the full measure of the difference 
between the two theorems as to generality. Hot only must we 
specialise by putting s = C, but s must become C in a very special 
way. In order to make this clear, let us take the particular case of 
Binet’s theorem which approximates as nearly as possible to the 
particular case of Cauchy’s given above. In the latter the deter- 
minants were of the 2nd order ; therefore to get the comparable case 
of Binet’s theorem we must put % -= 2. Again, since P in the 
particular case of Cauchy’s theorem was 10, we must for the same 
purpose put 
s=\0 
The result is 
and Cm.n = 19 , and m = 5 . 
to 
^ 21^22 
+ . . 
. - 1 - 
^ 10 , l ^ ho ,2 
^ 11^12 
^ 21^22 
^ 10 , 1 ^ 10.2 
1 ^ 11^13 
+ 
^ 21^3 
+ . . 
. + 
^ ho , 1 ^ 10.3 
1 611613 
^^21 ^23 
^ 10,1 ^ 10,3 
^ 11^12 
^lli^l2 
®10,1®'10.2 
^lOu/^10,2 
ttii a^3 
+ . . 
. . -f 
“ 10.3 
^10,1^10,3 
+ 
-f 
^24^25 , , 
0 
p 
'h^ 
) 1 1 ® 14^^15 
. , r^l0,4®10,5 
-+-.... "T ^ „ 
• 1^4 ^5 
KK\ ■■■■ 
^ 10.4 ^ 10,5 
nl/3ii/3i3 
1 ^10,4^10^5 
I <^11 ^12 • 
• • ^15 
+ 
^21 ^22 • • 
■ • <^25 
4- . . 
. -f- 
^lOU-^'10,2 • ' 
■ • ^10,5 1 i 
1 ^11 ^12 • 
• -^15 
^21 ^22 * ■ 
■ • ^25 
0,1 ^10,2 • ' 
■ • ^10,5 1 ^ 
1 ^11 «12 • 
• • ®'15 
j-f 
®21 “22 • • 
• “25 
1 
1 
®^10,1 ®^10,2 • • 
• «10,5 1 \ 
^11^2 • 
••A5I 
(^21(^22 • • 
. . 
^lOu/^10.2 • • 
’ • Ao,5 ' ^ 
the elements involved being 200 in number, and disposable in two 
sets of arrays — 
• • • ^15 ^21 <^22 • • • ^25 ^10,1 ^10,2 * * * <^10, 5 
^12 ' • • ^15 J ^21 ^22 • • • ^25 5 .... ^ 10-2 * * * ^ 10.5 > 
VOL. XV. 19/2/89 2 K 
