A DEVELOPMENT OF A PFAFFIAN HAVING A VACANT MINOR. 



55 



Here the first three frame-lines, when freed of the portions containing zero elements, 

 constitute a rectangular array of 3 rows and 7 columns, from which thirty-five (i.e., C 7)B ) 

 determinants of the 3rd order are formable ; and the initial proposition to be made good 

 is that every one of the 210 terms of these thirty-five determinants is a portion of a term 

 .of the given Pfaffian. To do this we have only got to put in mental contiguity for a 

 moment the definitions of a determinant and a Pfafnan : for each determinant term being 

 required to consist of three elements taken from the rectangular array referred to, no 

 two of which must belong to the same row or to the same column, complies with the 

 requirement regarding the first three elements needed to form part of a term of the 

 Pfaffian, viz., that they must be chosen from the first three frame-lines but that at the 

 same time no two of them must belong to the same frame-line. The next proposition is 

 that all the six terms of any particular one of the thirty-five determinants require the same 

 cofactor in order that terms of the Pfaffian may be produced. This is made clear by con- 

 sidering the fact that all of the six terms have their elements taken from the same set of 

 frame-lines, and that the remaining two elements in each case must be chosen from the 

 elements left when the said set has been deleted from the Pfaffian. Thus, if the 

 particular determinant were 



15 17 18 



25 27 28 



35 37 38 



each of its terms could only become a term of the Pfaffian by having annexed to it two 

 elements selected from those left when the 1st, 2nd, 3rd, 5th, 7th, 8th frame-lines of the 

 Pfaffian have been deleted — that is to say, from 



46 49 it 



69 6* 



9t. 



We are thus prepared to advance a third proposition derived from the previous two, viz., 

 that, apart from the question of sign, all the eighteen terms of each one of the products 

 of the form 



15 



17 



18 



. 46 



49 



it 



25 



27 



28 





69 



6t 



35 



37 



38 







9t 



are terms of the Pfaffian. This means that 35 x 18, i.e., 630, terms of the Pfaffian are 

 accounted for. Now the total number of terms in a Pfaffian of the 5th order is 1.3.5.7.9 ; 

 the number of these which will vanish when the element 12 vanishes is 1.3.5.7 ; the 

 additional number which will vanish when 13 vanishes is 1.3.5.7; and the additional 

 number which will vanish when 23 vanishes is 1.3.5.7. It follows, therefore, that the 

 total number of non- vanishing terms in the Pfaffian under discussion ought to be 



