A DEVELOPMENT OF A PFAFFIAN HAVING A VACANT MINOR. 



53 



For example, the Pfaffian of the 2nd order, 



12 13 14 



23 24 



34 



= 12.34 - 13.24 + 14.23 



(5) Instead of the elements being viewed as forming 2n— 1 rows of 2n — 1, 

 2n — 2, . . . , 1 elements respectively, and at the same time 2?i— 1 columns of 1, 

 2, . . . , 2n— 1 elements respectively, they may, without alteration of position, 

 be viewed as situated at the intersections of 2n frame-lines each containing 2ft — 1 

 elements, the r th frame-line being in every case made up of the r th row and the 

 (r— l) th column: and from this point of view the two integers used to specify 

 an element are the numbers of the two frame-lines in which the latter is situated. 

 This will be apparent from a glance at the following diagram of frame-lines for a 

 Pfaffian of the 3rd order : — 



I 



-> - 



12 ... 13 14 15 16 - 



23 - - - 24 



34 



25 • 



26 - 



35 36 



45 46 



56 



— > 1st frame-line 



2nd frame-line 



3rd frame-line 



- — > 4 th frame-line 



5 th frame-line 



6 th frame-line. 



It follows also that the rule for the formation of the terms is equivalent to a direc- 

 tion that all possible products of n elements are to be taken, no two elements in any 

 product being from the same frame-line. 



Thus, if in trying to form a term of the Pfaffian of the 3rd order whose elements 

 are a, b, c, .... , m, n, o, we took, to commence with, the element a in the first row, we 

 should thereby be debarred not only from taking anything else from this row (which is 

 the first frame-line) but also from the second row (which is a part of the second frame- 

 line), because a is an element in both, its place-name being 12. Our choice, conse- 

 quently, would then be from among the elements left after deletion of these two lines, 

 i.e., from 



