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L. On Skew Determinants. 

 By Thomas Muir, M.A., FM.S.E* 



1. ~TN 1855 it was proved by Brioschif, and, it is said, 

 J- simultaneously by Cayley, that any determinant of 

 even order is expressible as a Pfaffian. The restriction of the 

 statement to even-ordered determinants has since remained : 

 I propose now to show that it may be done away with. 



2. In the Quarterly Journal of Mathematics for this month 

 (p. 174) there is given a new form for the product of two de- 

 terminants ; viz. when the order is the third, we have 



«1 



«2 



«3 





ft 



ft 



ft 



X 



7i 



72 



73 





'l %2 X Z 



\ 1/2 y 3 



1 ~S 



-3 



1_ 



2 6 



«! + ^l « 2 + #2 «3 + #3 «3 — #3 « 2 ~^2 *f— #1 



ft+^i ft +y 2 ft+y 3 ft-3/3 ft-y 2 ft— yi 



7l+~l 72+^2 73 + ^3 73—^3 72—% 7l~ ^1 



7l— *1 72—^2 73 — ^3 73+^3 72+^2 7l+*l 



ft— yi ft— y a ft-y 3 ft+y 3 ft +3/2 ft+yi 



«1— #1 * 2 — # 2 «3~ #3 «3 + #3 *3 + #2* a l + #1 



If, in order to find from this an expression for the second 

 power of a determinant, we put x^ a- 2 , # s , . . . =a 1? a 2 , a 3 , . . . 

 the result is nugatory ; but, making the columns of the second 

 determinant the same as the rows of the first, we obtain 



«ift73l 2 = 2" 6 2*! « 2 + ft « 3 + 7l *3~7l «2— ft 



ft + « 2 2ft ft + y 2 ft-y 2 ft-« 2 

 7i + «3 72 + ft 273 y 2 — ft 7i— «3 



7i-<*3 72-ft 273 7 2 + ft y x + u z 



ft-«a ft-7 2 /3 3 + y 2 2ft /3 1 + « 2 



« 2 — ft «3~ 7l a 3 + 7l a 2 + A 2«j , 



By changing the signs of the elements in the last three rows 

 and reversing the order of the six rows, the determinant be- 

 comes skew with respect to its zero-diagonal ; and hence, on 



* Communicated by the Author. 

 t Crelle's Journal, hi, pp. 133-141. 



