676] 
265 
676. 
NOTE ON A THEOREM IN DETERMINANTS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 55—57.] 
It is well known that if 12, &c., denote the determinants formed with the matrix 
l| «» /3, V, $ ||, 
1 a, y, S' | 
then, identically, 
12.34 + 13.42 + 14.23 = 0. 
The proper proof of the theorem is obtained by remarking that we have 
a, 
/3, 7 > 
• 
a', 
¡3', y, 
a, 
/3, 7, 
s 
/ 
a, 
/3', y, 
S' 
as at once appears by subtracting the first and second lines from the third and 
fourth lines respectively; and, this being so, the development of the determinant 
gives the theorem. The theorem might, it is clear, have been obtained in four 
different forms according as in the determinant the missing terms were taken to be 
as above (8, 8'), or to be (a, a), (A, /3'), or (7, 7'); but the four results are equivalent 
to each other. 
There is obviously a like theorem for the sums of products of determinants 
formed with the matrix 
a , /3 , 
7 * 
8 , e , 
a , , 
7> 
S', 
r 
a", /3", 
// 
7 > 
8", e", 
r 
C. X. 
34