312 



DR THOMAS MUIR ON A PFAFFIAN IDENTITY, 



Taking the case where the given equation has six variables, and where the first form 

 of one of the denominators pertaining to the auxiliary equations is 



(BCE)(BFG) - (BCF)(BEG) + (BCG)(BEF) , 

 we find that the second form is 



where B\ B 1 



B 

 -C 



+ E 

 -F 

 + G 



c\ . 



qv E vi _ Qivpi + QivQV _ E v Qvi + giiipvi _ E iiiQv 



+ -p iv C vi - F" j E vi + F" ! G iv - G iv C v + G"'E V - G m F iv 



B v E vi_ B iv E vi + ■ 



+ pvgvi _ pii E vi + 



B v C vi_ B iii F vi + ■ 



+ F iii B vi _ F ii Qvi + 



B ivQvi_ B iii E vi + 



+ E iij B vi - E" C vi + 



B ivQv _ B iii E v + 



+ E m B T - E" C v + 



, G vi stand for 



SB 3B dC 8G 



Ta ' "36 ' ' da ' " ' ' ' dg 



respectively. The problem is thus suggested of justifying the use of one of these for 

 the other ; and with the setting of this problem the present paper originated. 



(2) By using the notation of the second form in writing the first form the latter 



becomes 



{BC iv - BE" - CB iY + CE" + EB m - EC" }{BF vi - BG V -FB vi + FG" + GB V - GF U } 

 - { BC V - BF m - CB V + OF" + FB" 1 - FC" }{BE vi - BG iv - EB vi + EG" + GB iv - GE" } 

 + {BC vi - BG"> - CB vi + CG" + GB m - GC" }{BE V - BF iv - EB V + EF" + FB iv - FE" } ; 



and it is readily seen that, whatever the relation between the two forms may be, it is 

 not dependent on the meaning here given to the indices, but that in fact it is a relation 

 connecting the elements of the five-by-six array 



B 





B 3 



B 4 B, 



B c 



C 



c 2 





c 4 c 5 



C i; 



E 



E 2 



E 3 



• E 5 



E 6 



F 



Fo 



F 3 



F 4 • 



F 6 



G 



G 2 



G 3 



G 4 G 5 





It is such relations, therefore, that have to be investigated. Before doing so, 

 however, the particular relation suggested by Pfaff may be established separately. 

 Changing the first form into 



(|BC 4 | + |CE, | + |EB 8 j)(|BF 6 | + | FG 2 1 + | GB 5 1) 

 - (| BC 6 | + |CF, | + | FB, |) (| BE, | 4- | EG 2 1 + | GB 4 1) 

 + (|BCJ + |CG,| + |GB 3 |)(|BE 5 | + !EF 2 | + | FB 4 |) 



and performing the multiplications indicated, we have an expression of twenty-seven 



