58 DR MUIR ON A DEVELOPMENT OF A PFAFFIAN HAVING A VACANT MINOR. 



and end with | 18 29 3t \ , the sign to be prefixed to any product is easily known 

 from that of the preceding product. 



(9) The number of different forms of this new development which are possible 

 in the case of a Pfafiian of the n th order is of course the number of partitions of 

 the integer n into two integers, the first of the latter corresponding to the order of 

 the determinant factors in the development, and the other to the order of the 

 Pfaffian cofactors. For example, in the case of the Pfaffian of the 5th order we shall 

 have the five developments 



(a) + 12J| 3456789* | - 13J| 2456789* | + (C 9>1 terms) , 



(/?) - | 13 24 |.'| 56789* | + | 13 25 |.'| 46789* | - (C 8;2 terms) , 



(y) - | 14 25 36 |.'| 789* | + | 14 25 37 |.'| 689* | - (C 7>8 terms) , 



(8) + | 15 26 37 48 |.9* - | 15 26 37 49 |.8* + (C 6>4 terms) , 



(e) +|162738495*| (C 5>5 term) , 



the parent Pfaffian containing no zero elements in the first case, 1 in the second, 

 1+2 in the third, 1 + 2 + 3 in the fourth, and 1+2 + 3 + 4 in the fifth. 



The single elements in the first development may be viewed as determinants of 

 the 1st order and in the fourth development as Pfaffians of that order. 



