Dr T. Muir on the Theory of Determinants. 
529 
and, as the new law is supposed to have been proved for deter- 
minants of the 3rd order, it follows that 
+ ~ ^1^2^41 "h ^ll^ 2 ^ 4 l} 
- — cfb^d^\ + c?il^2^3l} • 
Combining now by the original law the terms involving as a 
factor, the terms involving c^, and those involving d-^, we obtain 
I«1 ^3^4! = «ll^2^3^4l - M^2^3^4l + ^^11^2^41 " ^>2^4! 1 
and thus prove that the new law holds for determinants of the 
4th order. (vi. 6.) 
Cauchy’s development above referred to appears in the penulti- 
mate identity in the convenient form of one term afb^c^d^ 
followed by a square array of 9 terms. The form in Schweins’ is — 
\ I "V V / _ y+2'-l II ^n-x-\ an-a;+l 
K-i) ' K ' II ... 
Laplace’s expansion-theorem is next taken up. To prepare the 
way a theorem in permutations is first given, the enunciation being 
as follows ; — If from n different elements every ^permutation o/ q 
elements he formed, and every permutation o/ n-q elements ; and 
if each of the latter he appended to all such of the former as have no 
elements in common with it, all the permutations of the whole n 
elements will be obtained. Thus, if the permutations of 1 2 3 4 5, 
or say P (1 2 3 4 5), be wanted, w'e first take the permutations 
three at a time, viz., 
P(1 2 3), P(1 2 4), P(1 2 5), . . . . , P(3 4 5) 
where 1 2 3, 1 2 4, 1 2 5,. . ., 3 4 5 are the orderly arranged combina- 
tions of three elements ; secondly, we take the permutations two 
at a time, viz., 
P(12), P(13), P(14), , P(4 5); 
and, thirdly, we append each of the two permutations included in 
P(4 5) to each of the six included in P(1 2 3), each of the two in 
VOL. XV. 28/2/89 2 M 
