178 On Differential Equations 



of which equation a first integral is 



u= Cx' @ 



whence follow 



u' = Cx'\ u" = Cx''\ &c. - - (j,k. .) 



7. Our next object is to eliminate u from the co-efficient 

 of T', and for this purpose we deduce at once from (g) the 

 relation 



u _ X 



Q) 



moreover 



lb'' ■ ( iC" x'' , ^ 1 , . 



%L \ (xY (ccy 

 ^ {xy {xy 



(n) 



as we see from inspecting (i) and (j, k, . . ). Hence the co- 

 officient of Y' in (/) becomes, on substitution, 



and reducing this expression and equating the result to zero 

 we find, as the second condition of the proposed transfor- 

 mation. 



^^N 2 



an equation connecting x and t directly. 



8. It remains to be proved that (o) is reducible to a linear 

 difierential equation of the second order. To this end, let 



x' =2^ - - - - - - - 0^) 



and consequently 



x'' = p' = 2^' -p - ^ - - ' - (g) 



and also 



aj"^ = 2^ = 2f . p2 4 (jyy . iJ - - - . (r) 

 then (o) becomes, after substitutions, 



2^9 1/ + 2 Qy)2 — 3 {p'y + 3 r^^^ ^ q. 



