Axisymmetric Determinants. 



449 



consequently the whole of the cof actor has not been found. Trying 

 a result of Cayley's, namely, 



111 



N( six + \ly + Jz) = 



1 . z y 



1 z . x 

 1 y x 



and bearing in mind that for this determinant we may substitute 



we have 



P 



P 



(T Zf 



y° l 



Zf 



Xp l 



y* 1 



XJJ 



N( Ja 2 k + Jb 2 D + Vc 2 F) = 



<x 2 



= a 2 



b 2 c 2 



ad-b 2 af-c 2 



ad-b 2 . df- e 2 

 af-c 2 df-e 2 



a 2 b 2 c 2 



. F D 



F . A 



D A . 



a 2 b 2 c 2 



2a 2 ad af 



ad . A 



af A . 



a b 2 c 2 



2 d f 



d . A 



/ A . 



= a 2 



2 a 



a 



d b 2 



f c 2 



d f 



b 2 c 2 



. A 



A . 



It is thus seen that the missing cofactor is a 2 , and that we have 

 reached at one and the same time two theorems, namely, In any three- 

 line axisymmetric determinant A the cofactor of A in the norm of 

 (11) x/[TT] + (12) V[22] + (13) V[33] is 



(H) : 



and 



(11) (12) -(13) 



(12) (22) (23) 



(13) (23) (33) 



2 a 



d 



/ 



a 



b 2 



c 2 



d b 2 



m 



A 



f c 2 



A 





a b c 





a b 



-c 



b d e 



. 



b d 



e 



c e f 





-c e 



f 



(V.) 



(VI.) 



6. Since the right-hand member of (VI.) is not altered by per- 

 forming the cyclical substitutions 



a d f 

 d fa), 



e c b 

 c b e 



