218 



DR THOMAS MUIR ON 



the hypotenuse elements from (a r , a r+1 ) into a r a r+1 , the corresponding change on the 



right-hand being from (a ls a 2 , 



. . , Oz r , cx> r +i, • • • , cc 2n ) into 



(«H «2 a ') K+l> • ■ • > a 2n) • 



(20) Now as the Law of Complementaries holds in regard to Pfaffians as well as in 

 regard to determinants, let us apply it to find the Complementary of the general 

 theorem just reached. Instead of each element of the Pfaffian we have to substitute its 

 complementary minor ; and as the value of the Pfaffian is (a 1( a 2) . . . , a 2n ) the com- 

 plementary minor of any element is nothing else than its cofactor in (a 1; a 2 , • • • » a 2n)- 

 On the right-hand side the complementary minor of (a 1} a 2 , . . . , a 2n ) is 1, and therefore 

 we have to annex to it a power of (a 1} a 2 , . . . , a 2n ) of the same degree as the altered 

 Pfaffian on the left, viz., the degree n (2n — 2). 



I cof (a v a 2 ) cof a x a z cof a 1 a i . . . 



cof (a 2 , a 3 ) cof a 2 a 4 . . . 



cof (a 3 , a 4 ) ... 



We thus have the theorem 



= (ftj, Cl 2 , . . . , Ctzn) 



cof a^™ 

 cof a 2 a 2 „ 

 cof ct 3 a., n 



cof (a 2n _ u a 2n ) 



which is the generalisation surmised in § 10 to exist by reason of the two cases 

 established in § 9. 



(21) Now, however, that we have accomplished the purpose with which we started, 

 it is important to notice that the fundamental theorem of the whole is that given in 

 §19. Any other mode of establishing it is thus of interest; and the known quanti- 

 tative relation between Pfaffians and determinants suggests that by substituting 

 determinants for squares of Pfaffians, the theorem in determinants thus resulting may 

 be easy of proof, — in other words, that instead of proving, for example, that 



(a,b) 



ac 

 (b,c) 



ad 



bd 

 (c,d) 



= (a,b,c,d), 



we may seek to prove the identity resulting from this by squaring both sides, viz., the 



identity 



(a,b) ac ad a 1 



-(a,b) . (b,c) bd -1 b 1 



- ac - (b,c) . (c,d) . - 1 c 1 



-ad bd - (c,d) . - 1 d 



(22) Beginning then on the left-hand side, but taking a determinant of higher order 

 so as to leave no doubt of the generality of the process, viz., the determinant 





(a,b) 



ac 



ad 



ae 



«/ 



-(a,b) 





(^c) 



bd 



be 



¥ 



- ac 



- (^) 





(c,d) 



ce 



cf 



— ad 



- bd 



-(c,d) 





(d,e) 



df 



- ae 



- be 



- ce 



-(d,e) 



■ 



(e,f) 



- of 



- ¥ 



- '■/ 



- 'V 



-(*,/) 



. 



