314 



DR THOMAS MUIR ON A PFAFFIAN IDENTITY, 



where the dot below the 4 is used to indicate that, while the other integers vary their 

 position, the 4 remains unchanged throughout. It has now to be noted that the 

 constituent determinants of the aggregate are those of the three-by-four array 



which contain no zero elements : also that the said aggregate is expressible as the 

 difference of two Pfaffians, namely, 



where the first of the two is apparent in the original array, and the second is got from 



the first by substituting for a. 2 , a 3 , b s the elements conjugate to them in the said array. 



AVhen | a 1 b 2 c 3 | is axisymmetric the aggregate vanishes,* and when | a 1 b 2 c B | is skew 



the aggregate becomes t 



- 2 



-facts that are most readily verified by considering the bi-Pfaffian form. 

 The two-line minors of the determinant 



b. 



Co 



(/j '/., ^3 



which have no zero elements are C 4>2 in number, namely, 



I «3 & 4 I 



«2<\l 1 



1 h C 4 



a 2 d 3 | 



- 1 h \ c h 





1 eA 



The sum of the elements of the r th frame-line of this Pfafftan matrix is the aggregate 

 derivable from the r tb set of three rows of the determinant, and the sum of the 

 remaining elements of the matrix is the aggregate derivable from the r th set of three 

 columns. The sum of the aggregates derivable from the rows is thus the same as the 

 sum of the aggregates derivable from the columns, namely, 2 2, 

 (4) Similarly, the aggregate 



1 2 

 3 4 



a i b -A; I - I « 8 Me I + I «2 C 5<*6 



V:A l» 



which is representable by 



l 2 3 

 4 5 6 



■ Sitzungsb. d. k. Akad. d. IViss. (Berlin) (1882), pp. 821-824. 

 (• Proceedings Roy. Soe. Edinburgh (1!J00), xxiii. pp. 142-154. 



