A DEVELOPMENT OF A PFAFFIAN HAYING A VACANT MINOR. 57 



for the number of inversions found in the pairs obtainable by placing each integer 

 of the first group in front of each integer of the second group. In the former, 

 I(afiy . . .), the order in which we write the integers of the group is all-important ; in 

 the latter, l(a/3y . . . ; a'fi'y. . . ), the order is of no consequence so long as we do not 

 mix the two groups. 



With this notation a fundamental proposition regarding inverted-pairs, which bears 

 directly on the subject in hand, can be stated very simply, viz., 



I(a/3y . . . a'/3'y' . - . ) = I(a/3y . . . ) + I(a'ySV • • • ) 



+ I(a0y...; a'/3V'...), 



and it is immediately evident therefrom that whatever interchanges may be made in 

 the group a/3-y ... or in the group a/3'y. . . 



I(a(3y . . . a'/3'y' . . . ) - I(a/?y . . . ) - I(a'£y • • • ) 



will remain constant. 



As applied to the special groups of integers connected with the Pfaffian term 

 above-mentioned this ensures that 



1(1725384960 - 1(172538) - 1(4960 



will not alter by reason of any interchanges taking place in the group 172538 or in 

 the group 496L If in addition the integers 123 in the first group be excluded from 

 interchange, we shall have 



1(172538) = 1(7; 23) + 1(5; 2) + 1(758), 

 = 3 + 1(758), 



through all interchanges in 758. By substitution it therefore follows that the number 



1(1725384960 - 1(758) - 1(4960 



will remain constant while interchanges take place in 758 or in 496t ; and this is 

 equivalent to saying that the terms of the parent Pfaffian which are obtainable from 

 the product 



15 17 18 | . | 46 49 it 



25 27 28 69 6t 



35 37 38 I 9* 



are either all correctly or all incorrectly signed. 



If the said constant be even, the sign which ought to precede the product will of 

 course be + , in the other possibility - . As a matter of practice, however, the easiest 

 way of determining the sign to be placed in front of any product is to make the 

 sign such that one of the terms obtainable from the product shall be correctly signed, 

 and for this purpose the facts given in § 6 will be found useful. If the whole expan- 

 sion be wanted, and the products be arranged in the natural order — that is to say, 

 in such a way that the series of determinant factors shall begin with j 14 25 36 | 



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



