Mb. CAYLEY, on THE THEORY OF DETERMINANTS. 



79 



=.-Jf 



w A B 

 x' A' B' 



. (26). 



If the two sides of this equation are multiplied by the two sides of the equation (2), 

 written under the form 



,a/3 



a'y3' 



(27). 



The second side is reduced to 



-jr 



. a^ + lir,.. , a'^+13',,.. 



X K 



x' ■ K 



(28). 



= -Jf.(K)''-\U. (29). 



And hence 



iFU = jr.{Kuy-\u (30). 



And similarly 



Fiu = jr.(Kuy~'.u. (31). 



Also combining these with the equations (22), (23), 



... (32). 



YFU F^U U 



KFU K-JU KU 



It may be remarked here that if U, V are functions connected by the equation 



FU=cFV, or 1U= civ. (33). 



Then in general 



1 

 f7=en-i. V. (34). 



To prove this, observing that the first of the equations (33) may be written 



FU= F.(c!^.r), (35), 



we have 



I.FU^I.F.ic^.V), (36), 



or 



Jf.(,KU)''-\U=jr[K(c^'.r)]'-Kc^i.V (37). 



Or, if neither J, C nor {KU) vanish, this equation is of the form 



U=kV, (38). 



whence substituting in (33), 



fe"-' = c, (39). 



which demonstrates the equation (34) ; and this equation might be proved in like manner from 

 the second of the equations (3.3). If however, J=0, or / = 0, the above proof fails, and if 

 KU = 0, the proof also fails, unless at the same line w = 2. In all these cases probably, cer- 

 tainly in the case of KU = o, re 4=^ 2, the equation (34.) is not a necessary consequence of (33) ; 

 In fact, FU, or lU may be given, and yet U remain indeterminate. 



