( 49 ) 



V. — A Development of a Pfaffian having a Vacant Minor. 

 By Thomas Muir, LL.D. 



(Read March 19, 1900.) 



(l) The simplest possible case of the development referred to in the title is one 

 which is seen to follow instantly from the determinant definition of a Pfaffian. Thus, 

 by the said definition, 



1 



• • a 4 



«5 



a 6 



-J 











a 4 



a o «6 







• \ 



h 



h 







h 



h h 







C 4 



C 5 



°6 







c 4 



H C 6 









d 5 







- «4 - h 



-''4 



d o d 6 



■ e 6 





and therefore 



• 









- «v, - h 



~ % ~ <h, 



- e 6 • 



) 











= v/{- 



a 4 a 5 

 b, b. 



4 D 



a G 





- a i 



~ C 4 



- a- - a G 



- b. - b r 



- C r - C a 



b o 











= J\ a i h c e ! 2 . 















= ± 



a i 



h 5 C 6 1- 













the result reached being that a Pfaffian with certain zero elements is expressible as a 

 determinant. It should be noted that, as in many instances where this form of defini- 

 tion is used, there is an ambiguity as to sign : it is of less importance to note that here 

 the appropriate sign is — . 



(2) Another case may readily be established by using either as a definition or as a 

 proved theorem the recurrent law of formation which, in the original notation of Jacobi 

 and Cayley, is exemplified by the identity 



[123456] = 12[3456] - 13[2456] + 14[2356] 



- 15[2346] + 16[2345]. 



Thus when there is onlv one zero element and the Pfaffian is of the 3rd order, we have 



a,, 



a b 



«6 



= * 6 \ 





a 3 



a 4 



h 



h 







h 



h 



C 5 



C 6 









c i 



d, 



e 6 



+ c e| 





«4 

 &4 



a 5 







+ %\ 



h 





h 



d. 



- d R 



- b R \ a Q 





VOL. XL. PART I. (NO. 5). 



H 



