266 NOTE ON A THEOREM IN DETERMINANTS. [676 
viz. the theorem is obtained by development of the determinant in an identical 
equation, such as 
ß, V, 
0 = 
a , 
8 , ., 
a', 
ß', i, 
S', • 
a", 
ß", y"> 
S", •, . 
a , 
ß > 7 > 
S, e, ? 
a', 
ß', y, 
S', e, r 
a", 
ß", y > 
o , e , g 
but we thus obtain 15 results which 
If, for shortness, we write 
are not 
all equivalent. 
A = 123.456, 
- B = 124.356, 
- G= 125.346, 
27 = 126.345, 
- E = 134.256, 
— F = 135.246, : 
G = 136.245, 
-27 = 145.236, 
7 = 146.235, 
7 = 156.234, 
then the fifteen results are 
¿ + £- (7-7) = 0, 
A + B-E- J = 0, 
^_O+F-/=0, 
¿-£+£-# = 0, 
J. - E + F + G = 0, 
A-H- I - «7 = 0, 
B - C -G + 77 = 0, 
B-D-F + I= 0, 
5-2? + #+ 7 =0, 
B-F-G-J = 0, 
(7+27-27— 7 = 0, 
C - E + G + I =0, 
G -F-27- 7 = 0, 
2>_ J E + J F + tf = o, 
B-G- I - J = 0, 
which are all satisfied if only 
7 = . . +77 + 7 + 7, 
7 = 7*+<7 . .+7, 
(7=7* . + 27 . + 7, 
2) = . (2 . + 7 + 7, 
27 = 7 T +(r+27 + 7 + 7; 
and we thus have these five relations between the ten products of determinants 
71, 5, (7, 2), E, F, G, 27, 7, 7.