236 Report S.A.A. Advancement of Science. 



where /7i2, #13, ••• are the cofactors of ao, a^, ... in ^T). Similarly 

 it is found that 



Consequently we have 



i-=5 



1 = 1 

 where S stands for 



#12 (^'i-''2) - #13 ki— rts) + + #16 {A—ih) 



+ #23 (^■2-^3) -....- #6 {f2-K) 



— an expression which manifestly vanishes when P is axisynimetric. 

 The theorem is thus established. 



4. It will be observed that in the case of both theorems (§§2, 3) 

 each term of the aggregate SAr is expressible as the product of two 

 pfafftans, one of which is dependent only on the elements of 0, and 

 the other has a frame line of elements taken from P and the rest 

 from O. When n is odd (§2) the pfaffian which is free of the 

 elements of P is of lower order than the other ; when n is even it 

 is of the same order as the other and besides does not vary for 

 different values of /■. 



5. The portion of the theorem of §3 which relates to the vanishing 

 of the aggregate SA can be readily established, without previously 

 removing the factor \/ Q from Ar, by showing that the cof actor of 

 any one of P's elements differs only in sign from the cofactor of the 

 conjugate element. 



It is also worthy of note that the same result follows from the 

 general theorem* that if D^ denote the determinant formed by 

 replacing the 7'*^ roiv (not column) of any determinant O by the 

 ,-th ;-()2f, Qf ^i^Y other determinant P, then 



Sa„ = ;SA. 



for, in the special case before us, the two equals can be shown to 

 differ in sign and must therefore both vanish. 



* See Proc. Roy. Soc, Edinburgh, vol. xv., pp. 96-105. 



