522 



DR THOMAS MUIR ON VANISHING AGGREGATES 



a determinant of the n th order as the sum of n determinants of the same order. Since 

 the cofactors of a x , a 2 , b-^ , b 2 , P , Q manifestly cancel themselves, an alternative proof 

 will be obtained if we can show that the expression 







a 8 



«4 



H 









C l 



'A 



e i 







h 



h 



h 









*3 



^4 



h 



c l 



C 2 



c s 



C 4 



c 5 



- 



a 3 



C 2 



C 3 



C 4 



C 5 



d 1 



d 2 



C i 



d t 



<h 





a 4 



d 2 



C 4 



<** 



d, 



e i 



e 2 



H 



d 5 









a 5 



e 2 



C 5 



d, 



e 5 



cZ, a, 



c 2 



d 



P 2 



&3 



*4 



h 



( '3 



'•4 



c f> 



''4 



^4 



d, 



'-5 



^5 



e 5 



vanishes. This, however, leads to the consideration of the more general expression in 

 which the complementary of the minor of zero elements is not axisymmetric, say the 

 expression 



• B i B 2 B 3 



: ' C t C 2 C 3 



A 2 x x a x p 1 7l 



A. 2 X., a 2 (i 2 y 2 



A 3 X 3 a s f3 3 y 3 



A 2 A, 







Ci 



c 2 



c 3 



A 



x i 



a i 



A 



7i 



B 2 



X 2 



a 2 



& 



72 



B, 



a 3 Aj 73 



+ 



By Laplace's expansion-theorem this is clearly equal to 



B 1 x 2 /3 3 \ 



+ 





A: 



A 2 A 8 





A 



E 2 B 3 



C\ x x 



a i 



ft 7! 



C 2 x 2 



a. 



A 72 



C 3 x 3 



a 3 



A» 73 



B 2 C 3 



• 1 



Aj x 2 a 3 | 



A 2 C 3 



• 1 



Bj x 2 a 3 | 



A 2 B 3 



• 1 



Uj X 2 a 3 | , 



I ^C, I • I A^y, I - I B.C, 

 - I AjOj I • I B^y, I + I A X C 3 I 

 + I AjBg I ■ I CjiTsy, I - I A^g I • I Oj^jS, 



and therefore by recombination, the addition being now performed on columns, so to 



speak, 



A, B, C x 



A x B, C x • • 



A 2 B 2 C 2 • • 



A ! B x C x ^ yj 



cc 2 y 2 



A 2 B 2 C 2 



A 3 B 3 C 3 z 3 y 3 



A 3 B 3 C 3 



+ 



A 2 B 2 C, 



A x B 2 C 3 



^172 



+ 



A 3 Bg Cg 



• • 



A x B, G, 



x \ a l 



A 2 B 2 C 2 



Xn 0- 2 



A 3 B 3 C 3 



X 3 a 3 



A 1 B 2 C 3 1 • 



1 ^2 a 3 



A, B, C, *, ^ 



A 2 B 2 C 2 x 2 (3 2 



A 3 B 3 C 3 x 3 (5 3 



A^C, I ■ \x, (3 2 \ 

 = I A^Cj I ■ { - x 1 (/3 3 - y. 2 ) + x 2 (a 3 - 7l ) - x 3 (a 2 - ft) } . 

 It thus appears that the aggregate set for consideration has | A x B 2 C 3 | for a factor, 

 and that the cofactor — and therefore the whole aggregate — vanishes when | a x /3 2 y 3 j 

 is axisymmetric. 



This curious proposition led to an investigation which resulted in the following 

 series of theorems, the first of which is new only in form. 



(17) If to n — 2 given columns having n elements each there be appended the r th 

 column of a given determinant of the a. th order, and from the resulting array the r th 

 row be deleted, there being thus produced a, square array of the (n— l) th order the 

 determinant of vjhich is D r , then the aggregate 



2(-ir i D ( . 



will vanish when the given determinant is axisymmetric. 



