78 



Mr. CAYLEY, on THE THEORY OF DETERMINANTS. 



which latter equation is given by M. Cauchy in the Memoirs already quoted ; the proof in the 

 " Exercises,'" heing nearly the same with the above one of the more general equation (13). The 

 equation (l3) itself has been demonstrated by Jacobi, somewhat less directly. Consider now the 

 function FU, given by the equation (3). This may be expanded in the form 



J7f7= (r^ + s;,+ ..) [A.(Aa!+ Asc' + ..) + B (bx + v' iv' + . . ) + ..] + (15), 



(r'^ + s',; + . . ) [J'- (A.r + A'a;' + ..) + B'{bx + B'm'+ ..) + ..] + 



vhich may be written 



FU = x.iA^ + B,, + ..) + 

 w'. (A'^ + B' ,, + ..) + 



(16). 



By putting 



A = A . (eJ + vi'A'+ . . ) + B . (nS + B.'B'+ . . ) 

 B = A.(s^ + s'A'+ ..) +B.(sS + s'B'+ ..) 



(17). 



A'= a'. (rJ + b'^'+ . .) + b'. (rS + k'B'+..) 

 B' = A. (sA + s'A' + ..) + b'.{sB + s'B' + ..) 



Hence, 



KFU = 



A, B.. 

 A', B' 



A, B . . 



E, S . 



b', s' 



Or observing the equation (14), and writing 



A, B . 



a', b' 



R, S . 



r', s' 



= / 



= r 



This becomes 



Whence likewise 



Consider next the equation 

 IFU'- - 



KFU=J.f.{KUy-^ 



Kiv = j.r.{Kuy-'' 



(18). 



A, B. 

 A', ff 



(20). 

 ... (21). 



(22). 

 (23). 



b'™' 



■B.at + Kx + . , , sa; + s a? + 

 a^+b»;+..,A , B 



a'^ + b',, + . . , A' , B' 



I . 



. R S 



a A, B 

 x A', B' 



(19). 



(24). 



(25). 



