200 



DR THOMAS MUIR ON 



+ 



which is easily seen to be transformable into 



1 a 

 1 b 



1 c 



J.(OD«+A»B») - 



1 a 

 1 c 



1 b 

 1 d 



(B-D- + A-C") + L ^ 



1 b 

 1 c 



(B"O + A«0>). 



The aggregate under consideration can thus be changed into 



} a A.\ C A {(CD + ABVI A°B 2 C 4 D 6 | - (C 2 D 2 + A 2 B 2 )-| AOB^D 5 1 + (C 4 D 4 +A 4 B 4 )- 1 A^CD* 

 1 o I I 1 d | ( 



1 a 



1 c 



1 a 



1 d 



. I h I • | (BD+ AC) • | A°B 2 C 4 D 6 \ - (B 2 D 2 + A 2 C 2 ) • | AOB^D 5 | + (B*D 4 + A 4 C 4 ) • | A^C 2 !) 8 

 { (BC+AB) • | A°B 2 C 4 D 6 1 - (B 2 C 2 + A 2 B 2 ) . | A°B 1 C*D 8 1 + (B 4 C 4 + A 4 B 4 ) • | AOB^D 3 



1 c 



and this can only vanish identically when the three expressions enclosed in { } are 

 equal, for the cofactor of bd is the difference between the first and third, and the cofactor 

 of be is the difference between the first and the second. Now the sum of the said three 

 expressions is 



! A°B 2 C 4 D 6 j-2AB - | A^OD^-SA^* + | AOB^D 3 1 • 2A 4 B 4 , 



and this by the theorem for the multiplication of an alternant by a symmetric function 

 of its variables is found to be 



3|A°B 3 C 5 D 6 | + 3|AB 2 C 4 D 7 |. 

 Consequently each of the three is equal to 



| A°B 3 C 5 D 6 | + | AB 2 C 4 D 7 | * 

 The aggregate above reached can thus be changed into 



and therefore into 



1 a 

 1 b 





1 c 



i d r 



1 a 

 1 c 





1 b 

 1 d 



+ 



1 a 

 1 d 





1 b 

 1 c 



1 • (l A°B 3 C 6 D 6 | + | AB 2 C 4 D 7 |) 



1 a 



1 b 



1 e 



1 d 



1 a 



1 b 



1 c 



1 d 



A°B 3 C 5 D<5 | + | AB 2 C 4 D 7 



')■ 



where the vanishing factor is what each of the first factors in the original aggregate 

 becomes when we put A = B = C = D=1. 



* A direct proof that 



| A°B 2 C 4 D 6 | -(AB+CD) - | A°B'C 4 D 5 | • (A 2 B 2 + OD*) + | AoB'CFD 3 ! -(A 4 B 4 +C 4 D 4 ) 

 = | A°B 3 C 5 D 6 | + I A 1 B' 2 C 4 D 7 | 



is obtainable from the theorem above given regarding the sum of two fourth-order determinants, the parents being 

 the two determinants on the right, and the progeny the six determinants obtainable on the left by performing the 

 multiplications indicated, viz. : 



A A 3 A 6 A 7 



B B 3 B B B 7 



1 C 2 C 4 C 6 



1 D 2 D 4 D 6 



1 A 2 A 4 A 6 



1 B 2 B 4 B 6 



C C 3 C 6 C 7 



D D 3 D 6 D 7 



