AND RELATED VANISHING AGGREGATES OF DETERMINANT MINORS. 317 



and the related dotted aggregates 



Z I 1 2 3 I v i 1 2 3 

 I 4. 5 fi I. 2* 



The Pfaffian form of the first of these being 



Z I 1 2 3 I v I 1 2 3 

 I 4 5 6 | , ^ I 4 5 6 



a n - 



a 3 -c x 



a 4 - d 1 



a 5 - e a 



a 6- f l 



h~ C 2 



b 4 - d 2 



b 5 - e 2 



& 6 ~fi 





c 4 - a 3 



C 5 _ e 3 



C 6-f S 







d 5 - e 4 



d e -^ 



it is at once evident that it will vanish if each element of any one of its principal 

 minors vanishes.* If we take, for example, the principal minor which is the cofactor 

 of a 2 — b v namely, 



I C i — 3 C 5 ~ e 3 C 6~ *3 



the condition then is 



4 ' L 5 ' u 6 ' 



d 'o ~ e 4 d C, ~fi 



e e, ~f 5 



':,' ''i;> e u — "-% 1 e 3 ' ■' 3 » e 4 ' -M > /5 ! 



in other words, that | c z dfaf\ j shall be axisymmetric. There being fifteen principal 

 minors of the Pfaffian, and each one having corresponding to it a four-line coaxial minor 

 of the determinant | a 1 b 2 c s d 4 e 5 f 6 |, the result we thus reach is — The determinant aggregate 



2 4 5 6 I vanishes when anyone of the four -line coaxial minors of | a 1 b 2 c 3 d 4 e 5 f 6 1 is 



axisymmetric. 



The next aggregate, ^ . 1 „ , need not detain us, as we have already seen in § 5 



that it is equal to 



«3 



a i 



«5 



a 6 



b Z~ C 2 



b A - d 2 



b b - e. 2 



h -ft 





c A -d 3 



C 5" e 3 



c 6 -fi 







( h ~ e 4 



d e ~fi 



H-fs > 

 the dot over the 1 being equivalent to an injunction to strike out from the Pfaffian for 

 2 I 4 5 6 [ all the elements having 1 for a suffix. By mere inspection it is evident that 

 the conditions for evanescence are narrower than in the case of ^ L g » | , the result 

 now being — The determinant aggregate V \ \ vanishes when any one of the four- 



line coaxial minors of j b 2 c 3 d 4 e 5 f 6 1 is axisymmetric. 



When the single dotted line-number is below, that is to say, is a column number, 



* It would, of course, also vanish if each element in any one of its frame-lines were to vanish : that is to say, if 

 equivalence of conjugates were confined to any one row of the determinant | o, ] b<f i d i e ;t f a J , — for example, if 



a 2 1 a 3 i a i > a b 1 a & = "l > c l ) "l i e l j f\ ■ 



This suggests a new avenue of investigation, and an avenue of special interest, because the number of conditional 

 equations requisite for the evanescence of the aggregate may as here be less. Like previous writers, however, we are 

 confining ourselves to vanishing produced by axisymmetry of a whole determinant 



