Vanishing Aggregates. 
325 
16-17.] 
a b c 
k 
j a ) be the determinant formed similarly 
A(^ ft y ‘ ‘ k) that they did in the determinants from which they 
were taken. Let A( a $ Y 
\a b c 
from the columns. 
Theorem B . — Then 
2A 
a b c 
a 
ft y 
2ii; ? : 
where the number of determinants on each side is 
l n 
a 
7c ‘ 
Before considering the proof of this theorem, let us consider an example 
where n — 4, a = 2, b = c = 1, and therefore the number of determinants 
formed in each set is 
We have 
4 
1 0 1 
Li 
Tyi 
12 . 
R(n 11 a 22 & 33 c 44 ) + B(a n a 22 r 33 /j 44 ) + lfta n b 22 a 33 c 44 ) 
+ R(a ll c 22 a 33 & 44 ) + Pi(a n b 22 c 33 a 44 ) + L( <^11^22 ^33 ^44) 
+ B{b lx a 22 d 33 c 4 ft + B(c n a 22 a 33 b 44 ) + 11 (^ 11 ^ 22 ^ 33 ^ 44 ) 
-j- 1^(^11^22^33^41) ~t 1^(^11^22^33^44)'! l v ( c i 1^22^33^44) 
~ C(nn«22^33 C 44) "1 t>(^n^22^33^44) C(^n^22 tt 33 C 44) "t * * * ~! ^( C 11^ 22^33^4, t) 
Where for convenience R (a n a 22 b 33 c u ) stands for 
and C(a n a 22 b 33 c 44 ) for 
ffii 
a i2 
rt 13 
«14 
^21 
^92 
a 23 
ci 24 
^31 
^32 
^33 
^34 
C 41 
^42 
C 43 
C 44 
ffil 
^12 
^13 
C 14 
&2i 
cl 22 
^23 
C 24 
r-H 
cq 
e 
a 32 
^33 
C 34 
Ci 4l 
^42 
^43 
c 4\ 
The truth of this relation may be seen by expanding each determinant 
involved, by Laplace’s theorem, in terms of minors of the second order 
containing the a’s and their complementaries. Then it will be seen that 
the coefficient of any minor of the second order in the a’s on one side is 
equal to the coefficient of the same minor on the other side of the equation. 
Thus taking the coefficient of 
^11 ^12 ■ 
(%21 ^99 j 
