Determinants and Pfaffians. 



-37 



6. A little examination of the first form of the expression 

 denoted by S in §3 leads to an important theorem concerning 

 pfaffians only, viz. : — 



// P and Q be any 2n-linc pfaffians, and ffV be the pfaffian icJiosc 

 first frame-line is the v^^ frame-line of P and wJiose remaining portion 

 is tliat got by deleting the r^^ frame-line of Q, then 



2 (-I)^#r = O. 



o. 



The basis of the theorem is the fact that each element of P occurs 

 twice in '^{—lY'ffr and that its cofactor in one place differs only in 

 sign from its cofactor in the other. 



7. If now we reverse the positions of P and Q in the theorems 

 of §§2, 3 we arrive at analogous results which can be combined in 

 one enunciation, there being no need to distinguish the case where 

 // is even from that in which /; is odd. This single theorem may be 

 expressed as follows : — 



// P be any n-line determinant, Q a zero-axial sf^eio determinant 

 of the same order, and if Ar be the determinaid foimed by replacing 

 the rt^ column of P by the r^^ colnmn of Q, tlien 



r=i 



can be so tiansforined as to shoii) that it vanishes ivhen P is a.visvmmetri 

 For example, if P and be 



1 m 



-I . p 

 — /// — p 



—n — q —r 



