304 Proceedings of Royal Society of Edinburgh. [april 16 , 
Then multiplying this by the second factor, and neglecting such 
terms as 5450 where two indices are alike, and deleting pairs of 
terms which destroy each other, we find the result to be 
abed 
12 5 7 
13 4 7 
0 3 5 7 
4 0 5 6 
2 0 6 7 
3 2 4 6 
1 3 5 6, 
and thus know that the given determinant is equal to 
\a}b^c^d"\ + \a)-b^c'^d"\ + \a%‘^e^dd\ - \a%^c^d'^\ 
- _ Ydi^c'^d^l + \a}b^c^d^\ . 
(7) I have made the necessary calculations for all possible cases 
of the determinant 
\ a a‘^ a''F(5,c,(i) 
1 J> ]fY{c,d.,a) 
1 c d'¥(d,af) 
1 d d^ d F(a,&,c) 
where a^Y{h,c,d) is of the 9^^ degree. The following tabulation of 
the results needs no explanation except in regard to the notation 
employed for the determinants themselves. An example or two 
will readily make this clear. If 
Y{b,c,d) = bh^ + ¥d^ + c^dK^ 
the above determinants would be denoted by 
A(0 1 2r; 4,4); 
and if the symmetric function were %h"^cPd} the symbolism for the 
determinant would be 
A(0 1 2 r; 5, 3, 1). 
