PROFESSOE CATLEY ON THE THEOET OF MATEICES. 



31 



15. Lemma. The determinant 



V= a, b, c, d 



e , f, g, h 



i , j, k, I 



m, n, 0, J) 



may be expressed, and that in two different ways, as a Pfaffian. 



16. In fact multiplying the detei'minant into itself thus, 



V^= 



we find 



V==(a, b, c, d) 



(e , f, g, h) 



{i , j, k, I) 



(m, n, 0, p) 



a, b, c, d 



e » fy g^ ^ 



i , J, k, I 



m, n, 0, p 



tr. 



d, c, — b, — a 



hy g^ -fy -^ 



I, k, —J, —i 



p, 0, —n, —m 



(d, e, —b, —a), (A, g, —f, -e), (I, k, —j, —i), (p, o, — «, -m) 



^in ^125 ^135 °u 

 Sjn ^22, 523, °U 



^415 ^42 5 *43 5 ^44 



viz. we have Sii=(a, b, c, d)(d, c, —b, —a), Sii = {a, b, c, djli, g, —f, —e), &c.: we see 

 at once that Si, = 0, Su+S2, = 0, «&c., viz. the determinant in 5 is a skew determinant, 

 that is, the square of a Pfaffian. We have therefore 



V =(Sli! 534-1-5,3842 + 5,4523) 5 



or extracting the square root of each side, and determining the sign by a comparison of 

 any single term, we have ^ _l , _i_ 



V 5,2 534-1- 5,3 *'42 + ^14 *23 5 



which is one of the required forms of V. 

 17. And in the same manner 



V^=: tr. 



which is equal to the determinant 



'm 'l25 ''135 *14 



^21 5 ^22 5 ^23 ) ^24 



''31) ''32 5 ''33 5 ^^34 



^415 ^42 5 ^43 5 ^^44 



=(a, e, i, m) 

 (*5 /, i, n) 

 (c, g, k, 0) 

 (d, h, I, p) 



