212 
Proceedings of the Royal Society of Edinburgh. [Sess. 
in onr set of operations the multipliers of the columns had all had the 
suffix 2, we should similarly have proved | a 1 b 2> c^d b | to be a factor ; that if the 
suffix had been 3, the factor reached would have been | a-p 2 c^d b | ; and so on. 
Now, as the product of the five factors thus obtained is of the 20th degree 
in the elements, and the original determinant is also of the 20th degree, 
we have only to ascertain the connecting factor of degree 0. To do this, 
we note that in the special case where the matrix is 
a x a 2 
\ • ^3 
C 1 • • C 4 • 
d 5 
the original determinant reduces to its diagonal term, but that, although 
each of the set of five determinants reduces to one term, this term cannot 
be made the diagonal term in every case without effecting certain row- 
transpositions which necessitate 1 + 2 + 3 changes of sign. The connecting 
sign-factor is thus seen to be ( — l) 6 when n = 4>, and generally to be 
( — (I.) 
(3) The fact that the determinant under consideration in § 2 was 
resolved into two factors, and that only one of the two was utilised for 
evaluation purposes, enables us to give an interesting theorem in reference 
to the second factor. For, the result of the first resolution being, say, 
A = a 1 b 1 c 1 d 1 • | « 2 ^ 3 c 4 c? 5 | • \-cgW, 
\ a i u i c i a i) 
and the final result being 
A = I I • I «1 V 3 d 5 I • I ffiVA I • I a A c A I • I «2 6 3 c A I > 
it follows that 
W = (a-fi^d^f • | eq b 2 c^d^ | • | a^^d^ | • | a 1 b 2 c 4 t d 5 | • | a 4 6 3 c 4 (i 5 I (IT) 
The formation of the compound determinant W is peculiar. If the 
originating array consist of n rows and n + 1 columns, the order of W is 
\n(n— 1). The two-line determinants which go to form it are all the 
two-line minors of the array which have their first column coincident with 
the first column of the array : they therefore themselves form an array of 
\n(n — 1) rows and n columns. By combining in pairs the n elements of 
each of these \n{n — 1) rows we obtain \n{n — 1) products to be the 
elements of the corresponding row of W. It is this want of similarity 
in the mode of equalising the length and breadth of the W matrix which 
accounts for the absence of symmetry in the result. 
