SOME IDENTITIES CONNECTED WITH ALTERNANTS. 



189 



any term of the determinant on the left, — the term —ajb. 2 c x say, — the corresponding 

 term of each of the determinants on the right is —a^b 2 c x multiplied by a term of the 

 symmetric function, the multiplying term being different for every determinant.* 



Applying this theorem to the left-hand member of Cayley's first identity, and for 

 shortness' sake writing only the first line of each determinant, we have 



A 2 «A A 



which, as the first and last terms combine into one, and the second and fourth terms 

 vanish, reduces to 



+ 



| A 2 a A 2 | 



+ 



| 1 aA 2 A 2 | 



+ | A aA 2 A 



+ 



| A a A 3 | 



+ 



I 1 aA A 3 | 



+ 2 j A aA A 2 



- | A 2 aA A | + | 1 aA 2 A 2 

 Similarly the right-hand side becomes 



+ | A a A 3 1 + ! 1 «A A 3 



A 2 a A 2 



+ 

 + 



1 aA 2 A 2 

 A a A 3 



+ 



+ 



1 a A 4 

 A aA A 2 



+ 



laA A 3 

 1 a A i 



which in like manner reduces to 



I 1 aA 2 A 2 | + | 1 aA A 3 



+ j A a A 3 I + I A aA A 2 



And as these two results are the same, the identity is established. 



(3) The second identity is readily proved in the same way, both sides being equal to 



| A 2 a aA 2 | + j A a aA 3 | + j 1 aA aA 3 | + | A aA aA 2 | . 



It is however essentially the same identity as the first, as may be seen on writing 

 a" 1 , 6" 1 , c" 1 for a, b, c respectively in either of them. 



(4) The same method of course suffices for the proving of the third and fourth 



* Another theorem on the same subject may be illustrated by the same example, viz. : 



<h a 2 



b l b 2 



«3 

 *>3 



Co 



( A^B + A 2 C + B 2 A + B 2 C + C 2 A + C 2 B) 







1 2 

 C l C 2 



6 



a 3 A 2 B 

 6 3 B 2 C 

 c 3 C 2 A 



4 



a t a 2 a 3 A'-'C 

 6 X 6 2 & 3 B 2 A 



Ci Co Co kj"ij 



+ 



a i 

 h 

 c i 



a 2 

 h 



a 3 B 2 A 

 6 3 C 2 B 

 c 3 A 2 C 



«1 «2 



b i h 



C l C 2 



a 3 E 

 6 3 C 

 c 3 A 



2 C 

 2 A 

 2 B 



+ 



a t a 2 a 3 C 2 A 

 &! b 2 6 3 A 2 B 

 *i H %B 2 C 



+ 



a i 

 b x 



c i 



a 2 

 h 

 c 2 



a 3 C 2 B 

 6 3 A 2 C 

 c 3 B 2 A 



Here only one row or column of the original determinant is multiplied, the multipliers being complete terms of the 

 symmetric function. Each multiplying term, it will be observed, is used three times, and occurs in a different position 

 every time ; for example, the cofactor of A 2 B on the right is 



h h 



which is equal to | a t b 2 c 3 1, as it should be. 



«1 <*2 



b l b 2 



+ 



