and on Go-resolvents. 179 



and; on reduction, 



^Vlf—{j>'f+^r2J^ = 0- - - - (s) 



Now divide this result by p^ and we have 



or 



^iii^)+(?r-^«-« - -(^0 



2^ J V p 

 and if we make 



^ = 2v - - - - ^ (v) 



then (u) becomes, on dividing by 4 

 dv . 0.3 



cZa. + ^'+i^' = ^ - - - ('') 



whence also 





'''''' u>^ + l^-]-'^ - 



- («=) 



or, making e^"''' = iv, 



- (2/) 



cFw ,3 



- (^) 



a linear diiferential equation of the second order. 

 9. From (y) we deduce 



1 d w , ^ 



IV a X ^ ^ 



and, combining this with (y) we find, on integration, &c., 



p = C2 ^v^ . . - . (ah) 

 whence, by (p), 



fLi = 1 = -J_ 



dx p G2W^ " V ; 



or 



We find also, combining (i) and (p) and (a6) that 



w — C G^ 10^ - - - - (ae) 

 which two constants are of course equivalent to one only. 



N 2 



