of Differential Equations. 533 



which are all of the Clairaut type ; while the example (4) is due 

 to the combination of the equation 



y —px = a V\ + ja', 

 also of the Clairaut type, with its singular solution 



x'^ + y^=a^. 



The above examples, when reduced to a symmetrical form, 

 would appear thus : — 



/,M / du du\du afdv>? _ 



,n\i ( du , duXfdu du\ fdu\^ 



(^) Vdi^yTy)\d-x-Ty) = ''\Ty)' 



.„, / du du\^_/du\^_fduY (Taylor's 



^**^ rdx'^yd^J~{dyJ \di) ' equation) 



the general solutions of which are, as we have seen, severally 



(«^- + /9y)(«-/3) = «/3^ 



{ux + fiyY{a^ + ^)=a\ 

 and the singular solutions 



(a; + y)2_4ay = 0, 



y'^—x'^ = \, (Taylor's solution) 



2 2 2 



x^->ry^-=a^. 



It is worthy of remark, too, that in the case of the examples 

 numbered (5) and (7) in the same collection, namely, 



jj^-^yp + x = 0, 



y'^p^ — 2xyp + ax-]-by = 0, 



the method when applied seems to fail, inasmuch as the results 

 obtained, respectively, 



y^ — 4x=:0, 



ax + by — ,^^ = 0, 



cannot be regarded as any solutions at all, inasmuch as they do 

 not satisfy the corresponding equations at all. 



Now it appears pretty evident that this result might have been 

 anticipated beforehand, and that the theory is deficient in not 



