Me. CAYLEY, on THE THEORY OF DETERMINANTS. 



83 



K{U+U).^U-KU.l (U+ U) = 



IK{ U+U) \^- 

 \ KU I 



\'E[Ap-^ Ba..) (a^ + B,i) + {A',j + B'a-..) (a'^ + b'^ . . ) . . ] x 



[(Aa + A'a . . ) {vix + rV . . ) + {Ba + B'a . . ) (s,r + sV ..)..] j 



K {U + U) FU - KU . F {U + U) 

 IK(U+U) 



X \^{Aa + A'a . . ) {ax + ax . . ) + {Ba + B'a' . . ) {bw + bV . . ) . . ]| , 



which however are not true, except for n = 2, on account of the equation (57). In the case 

 of n = 2, these equations become 



K{U + U).'IU- KU.T{U +U) = 



[{Ap + Ba..){Al+Br,..) + (A'fj + B'a..){A'l+ b'v ..)+■•] x 



\{Aa + A'a ..) {nw + e V . . ) + {Ba + B'a . .) {s.v + s V . . ) + . .] (68), 



K{U+U)FU -KU F{U+U) = 



[{Afj + Ba..) (r^+ 8,,..)+ {A'p+ 5'cr..)(R'^+ s'v ..) + ..] x 



[{Aa + A'a ..) {aw + aV..) + {Ba + B'a ..) (b.i; + bV + ..)+.] {6d), 



and it is remarkable that these equations ((68), {69)) are true whatever be the value of (w), 

 provided 2 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 {6g) 

 alone need be proved, (68) following immediately when this is done. 



I premise by noticing the following general property of determinants. The function 



a + 2joa, 



/3 + So-fi, 

 a + 2joa', /3' + 2cr'a, 



(70), 



(where 2joa = ^iffli + p^'^z ••• + Ps^s)i contains no term whose dimension in the quantities a, as'... , 

 or in the other quantities p, cr... , is higher than s. (Of course if the order of the determi- 

 nant 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. 



2pa, Sera 

 S/3 a, 2(7 a 



(71), 



whenever («) is less than the number expressing the order of the determinant. Hence in the 

 formula (70), if 2 contain a single term only, the first side of the equation is linear inn, cr..., 

 and also in a, a',.., i.e. it consists of a term independent of all these quantities, and a second 

 term linear in the products pa, pa'... era, ad.... This is therefore the form of K {U + U) ... 

 Consider the several equations 



K = KU= Aa + Bji + (72). 



= A'a + B'(i' + + 



It is easy to deduce 



K^ = K.{U+U) = KU+ Apa + Baa + 



+ A' pa' + B'a a' + 



. (73). 



LS 



