254 Proceedings of Royal Society of Edinburgh. [sess. 
= 1, the determinant resolves itself into binominal factors, which 
are got by subtracting each of the other variables from 1. 
Writing 3 for ~ — 1, and subtracting and adding we 
have the still more pleasing result 
D(a 1 ,a 2 ,a 3 ,q 4 ,a 5 ) = g ga a g_ + 2W,3 4 . (S 6 ) 
a 1 a 2 a 3 a 4 a 5 
(8) If we diminish each row of D(a v a 2 ,a^a A ,a 5 )j a 1 a 2 a 3 a 4 a 5 
by the row which follows it, and thereafter diminish each column 
by the column which follows it, the determinant resulting is an 
axisymmetric continuant, the identity being 
D(a 1 ,a 2> a 3 ,a 4 ,a 5 ) 
a l a 2 a 3 a 4 a 5 
d 1 + d 2 d 2 
d 2 d 2 + 0 3 S3 
0 3 0 3 + 0 4 0 4 
04 0 4 + 0 5 0 5 
05 % + l • 
(9) Turning now to N(a l3 a 2 ,a 3 ,a 4 ,a 5 ) we note first that it is 
obtainable from D(a 1 ,a 2 ,a 3 ,a 4 ,a 5 ) by deleting the first column of 
the latter and substituting cq , a 2 , a 3 , a 4 , a 5 . The first row and 
first column of N are thus identical, and cq, instead of being 
as in D in every place except 1,1 , occurs in that place only. This 
suggests the partition of N(cq, a 2 ,a 3 ,a 4 ,a 5 ) into the aggregate of 
terms containing cq and the aggregate of terms independent of cq, 
the resulting identity being 
N(cq,a 2 ,a s ,a 4 ,a 5 ) 
aiD(a 2 ,a 3 ,a 4 , a 5 ) + 
a 2 
a 3 
a 4 
a 5 
a 2 
1 
a 3 
a 4 
a 5 
a 3 
a 2 
1 
a 4 
a 5 
a 4 
a 2 
a 3 
1 
a 5 
a 5 
a 2 
a 3 
a 4 
1 
Now the subsidiary determinant on the extreme right can, by 
the process of interchanging any two rows except the first, and 
subsequently interchanging the corresponding columns, be shown to 
be a symmetrical function of a 2 , a 3 , a 4 , a 5 , — say /( a 2 ,a 3 ,a 4 ,a 5 ). 
It follows therefore that both the cofactor of cq in N and the 
