122 Mr. R. Moon on the Theory of 



which, taking alternately the upper and lower signs, constitutes 

 a pair of equations of condition which must be satisfied when 

 /is susceptible of both the values (9). 



Moreover, since if (9) satisfies (7) the latter will be satisfied 



we have the further equations of condition, 



dp \ 2p ) 



V+-v/V ! 





2p* ctv\ 2p 



which reduce to the following, viz. 



Hence if (10) hold, we must have 



V 2 + 4R^ = 0, 



a conclusion which may be rejected on account of defect in 

 generality*. Therefore when equations (10) hold, (9) must 

 reduce itself to 



and we shall thus have for / four values in all, viz. those given 

 by the last equation together with those previously found, viz. 

 co l} co 2 . 



Now it will be remembered that the original integral which we 

 assumed for (2) was (2 a), or, which is the same thing, 



f{zytpv)=x{f a (%ytpv)\ (12) 



It is clear, however, that this last equation may be put under the 

 form 



fa{%ytpv)=X_ l {f(3sytpv)} i 



where x~ i denotes an arbitrary function; and if we treat this 

 equation in the same way in which we have treated (12), we 

 shall arrive at precisely the same formulae for the determination 



* The grounds upon which I rest this conclusion will hereafter be more 

 distinctly pointed out. 



