12 
Proceedings of Royal Society of Edinburgh. [sess. 
with the elements of the other, and a self-conjugate term is one 
into which there might enter self-conjugate elements and pairs of 
conjugate elements. 
6. Developing any determinant 
11 
12 
13 
14 
15 
21 
22 
23 
24 
25 
31 
32 
33 
34 
35 
41 
42 
43 
44 
45 
51 
52 
53 
54 
55 
in terms of binary products of the elements of the first row and of 
the first column, we have the unique product 
11 
22 
23 
24 
25 
32 
33 
34 
35 
42 
43 
44 
45 
52 
53 
54 
55 
1 products of the type 
12. 
21 
33 
34 
35 
' 
43 
44 
45 
53 
54 
55 
and (n- 1) (n- 2) products of the type 
12. 31 
23 
24 
25 
43 
44 
45 
53 
54 
55 
Now each of the products of the first two types consists of two 
factors, one manifestly self-conjugate, and the other a determinant 
of a lower order than the original determinant, but otherwise quite 
similar to it. On the other hand, in the third typical product, the 
first factor 12.31 is not self-conjugate, and the second factor cannot 
furnish the conjugate elements 21, 13. These considerations give 
us at once the equation 
■U M = U M _ 1 + (rc-l)U n _ 2 . 
