40 Transactions of the South African Philosophical Society. 



it a Pfaffian of the third order, so that the terms of the second kind 

 just given would be followed by 



a 2 a 3 a^ a. a 6 



6 3 b t b s h 



c 4 c. c 6 



d, d 



cof ; 



and so on. 



It is also worth noting that there is an interesting alternative 

 form which may be substituted for the first three terms in (E 2 ), 

 (E 3 ), ..., viz., 



CI/ 2 Cv-j CJL. 



. compl - 



6^2 Cvo tX- 



^3 b 5 



. compl + 



a 2 a 3 a 2n 

 b 3 b 2n 



.compl. 



7. When the identity (E T ) is applied to a Pfaffian of the 2nd 

 degree, (E 2 ) to a Pfaffian of the 3rd degree, and so on, the 

 development in each case ends with only one determinant under 

 the sign of summation : that is to say, we have 



Lb 2 (Xf*y Ct. 



b 3 Z> 4 



— a 2 .c^ 



(xo 



and the other special cases marked (\ 2 ), (X 3 ), (\ 4 ) above, &c. To 

 all except the first two of these there attaches, of course, the 

 blemish attaching to (E 3 ), (E ), .... It is greatly magnified, however, 

 if, instead of viewing the \ identities as giving the development 

 of the single Pfaffian on the left, we transform them so as to 

 present an equivalent for the solitary determinant on the right. * 

 Not only so, but the blemish then attaches to (\ x ) and (\ 2 ) also. 

 Thus, taking the case of (\ 2 ), which then becomes 



a 4 a. a 6 



K b] b 6 



= | a 2 a 3 a 4 a. a 6 

 b, b. b. b 6 



3 4 5° 



- a 2 \ 



C 4 C S C (> 



d. d f 



+ ., a 3 1 6 4 b. b 6 

 d s d 6 



£4 £5 ^6 



£4 C- c 6 



d 5 d 6 



e 6 





e 6 



Co 



- b 3 \ a 4 a 5 a 6 



d 5 d 6 



e 6 



* This is the way in which the identities are viewed in a paper by Mr. J. Brill, 

 which has just appeared in the Proceedings of the London Math. Soc. (see Vol. I. 

 of Second Series, pp. 103-111). The subject of the paper, it may also be men- 

 tioned, might well be overlooked, as the title under which it appears is " On 

 the Minors of a Skew-symmetrical Determinant," whereas the only theorem 

 contained in it is that here (§ 7) illustrated. 



