392 Mr. T. Muir on Skeiv Determinants. 



extracting the square root of both sides, there results 



= I M* 2 -ft) t(*3-ri) K» 3 +7i) K* 2 +ft) 



#(/33-y 2 ) t(ft+7 2 ) ft i(ft+* 2 ) 



7a &7 2 +ft) K7i+«3) 



2(^3-72) 2< a 3-7l) 



K« 2 -ft) 

 Similarly we have the determinant of the fourth order 

 I a ife73^4 [ equal to 



"l 



«2 



«3 



ft 



ft 



ft 



7i 



72 



7s 



(I.) 



U(« 2 -ft) i(-a-ri) «-4-*i) iK+^i) 



K«3+7l) *(«2+ft) 



Mft-72) *(ft-* a ) *(&+*,) 



*(ft + 7 a ) ft i(* 2 +ft) 



i(74-^s) #74+*.) 



7 3 *(ft+7 2 ) i(* 3 +7i) 



^4 



*fa + *.) i(ft + * 2 ) K*4 + *l) 





to*-'.) *(ft-* a ) iK-^i) 





i(ft-7 2 ) iC-s-rO 



*(*2-ft) 



and it is seen to be generally true that any determinant is ex- 

 pressible as a Pfaffian. 



The Pfaffian, it may be noted, is symmetric with respect to 

 its diagonal or axis. 



3. Brioschi's result for even-ordered determinants being 

 different in character from that obtained in the above way, 

 there is thus brought to light an identity between two Pfaffians 

 > — an identity leading to the theorem that a symmetric Pfaffian 

 of the 2nth order is expressible as a Pfaffian of the nth order, 



4. Again, as the product of two determinants is expressible 

 as a determinant in a way different from that of § 2, and each 

 of the said three determinants is expressible as a symmetric 

 Pfaffian, it follows that the product of tivo symmetric Pfaffians 

 is expressible as a symmetric Pfaffian. 



Other examples besides this of the application of our fun- 

 damental theorem will readily present themselves. 



5. By skew or symmetric determinants are usually meant 

 determinants which are skew or symmetric with respect to the 

 principal diagonal, "skew" and "symmetric" being con- 

 trasted terms. There is, however, possible another kind of 

 symmetry, and therefore also another kind of skewness, viz. 

 with respect to the centre, or point of intersection of the two 

 diagonals. 



In a centre-skew determinant, the wth row from the be- 

 ginning with its elements reversed in order and their signs all 

 changed, becomes the mth row from the end. 



A centre-skew determinant of the 2nth order is equal in 

 magnitude to the corresponding centro-symmetric determi- 



(II.) 



