224 



DR THOMAS MUIR ON THE 



Each term, it will be observed, is constructed from two distinct sources, viz., the 

 axisymmetric determinant and the quasi-Pfaffian. From the first source we have 



in the first term the determinant itself, 



in the second term, viz., D the array of its secondary minors, 



in the third term, viz., E the array of its quaternary minors, 



the last of these being merely the elements of the determinant by reason of the fact 

 that the order of the determinant is the 5th. Now there is no difficulty in writing all 

 these in the same way if we employ the so-called umbral notation throughout. They 

 are in fact 



1 

 1 



2 3 

 2 3 



4 5 



4 5 



, 





12 3 

 12 3 



, 



12 3 



12 4 



, (10 2 in number) 



1 

 1 



> 



1 

 2 







Again, from the second source we have 



in the third term, viz., E the primary minors of the Pfaffian, 



in the second term, viz., D the tertiary minors , 



in the first term nothing, 



the last involving no anomaly because the quasi-Pfaffian we are dealing with has only 

 jive frame-lines. Now, if we adopt for this quasi-Pfaffian an umbral notation analogous 

 to the preceding, viz., 



'| 1 2 3 4 5 | 



it will be found possible to represent its minors in a fashion closely resembling that for 

 the minors of the determinant. In fact, its primary minors will be 



l| 2 3 4 5 | , i| 1 3 4 5 | , 

 and its tertiary minors 



1 4 6 I 1 . 'I 3 5 | , '|3 4 

 The three terms of our development thus are 



12 3 4 5 

 12 3 4 5 



(5 in number) 



(10 in number) 



4 5 



3 5 



3 4 



12 3 



12 3 



12 4 



i 



12 3 

 12 4 



1 



12 3 



12 5 





12 3 







4 5 



3 5 



