12] 
ON THE THEORY OF DETERMINANTS 
Comparing these latter forms with the two equivalent quantities forming the first side 
of (53), and observing (33), (34), it would appear at first sight that 
K(U+ U).VU-KU.V(U + U) 
— ^ ^I£U ~) ^ [(Ap + Bcr ...) (A* + B?7...) + (A'p + B'a...) (a'£ + B't) .] 
x [(Aa + A'a'...) (rb + rV...) + (Ba + B'a'...) (s« + sV 
K(U +U). FU — KU. F(U + U) 
'K(U + U)\S 
KU 
X {[(Ap + Bcr...) (Rf + S77...) + (A'p + B'a...) (r'£ + sV..) ...] 
x [(Va + ^ V ...) (ax + aV ...) + (.Bci + B'a'...) (b# + bV ... 
which however are not true, except for w = 2, on account of the equation (57). In 
the case of n = 2, these equations become 
K(U+U). VU-KU.V(U + U) 
= [(Ap + Ba ...) (a£ + B?7 ...) + (J/p + 5'ct ...)(A / £ + B / 77 ...)+...] 
x [(Aa + A'a! ...) (rt + rV ...) + (Ba + B'a'...) (s# + sV ...)...] 
K(U+ U) FU — KU. F(U + U) 
= [(Ap + Ba ...) (r£ + St; ...) + (A p + B a ...) (R £ + ...) ...] 
x [(Aa + A'a'...) (ax + a'x' ...) + (Ba + B'a'...) (bx + bV + ...)...] 
and it is remarkable that these equations ((68), (69)) are true whatever be the value 
of n, provided X contains a single term only. The demonstration of this theorem is 
somewhat tedious, but it may perhaps be as well to give it at full length. It is 
obvious that the equation (69) alone need be proved, (68) following immediately when 
this is done. 
I premise by noticing the following general property of determinants. The function 
a +Xp a, /3 + 5V a,... 
a' + %p a', ¡3' 4- 2a'a, 
(where Xpa = p l a l + p 2 a 2 ... + p s &s)> contains no term whose dimension in the quantities 
a, a' ..., or in the other quantities p, a..., is higher than s. (Of course if the order 
of the determinant be less than s or equal to it. this number becomes the limit of 
the dimension of any term in a, a'... or p, a..., and the theorem is useless.) This 
is easily proved by means of a well-known theorem, 
a, %aa,... 
C. 
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