138 ON THE SOLUTION OF 



I shall next consider a celebrated problem, first proposed 

 by Euler, in the Petersburg!! memoirs. 



In a certain class of curves, the square of any normal ex- 

 ceeds the square of the ordinate drawn from its foot by a 

 certain quantity a. 



Let y 2 = tyx be the equation of the curve. The subnormal 

 is therefore fafrx, and the equation of the problem conse- 

 quently is 



+ (x + W*)=i + lWx)*-* ............ (16). 



Differentiating this, we get 



or 



The first of these two equations leads to the singular solu- 

 tions. In order to solve it, let 



x -f fflx %x, 

 then %*x %x x + # = 0. 

 Hence by the transformation already noticed, 



whence U K 



where Pz and P^z are functions of z, which remain unchanged 

 when z increases by unity ; 



therefore u g+l = Pz + (z + 1 ) P^z, 

 Hence we have 



* t for the required solution. 

 x = Pz + zPf) 



therefore ydy = Pf (Pz + P t z + zP^z) dz ; 

 and integrating by parts, we get 



f = Pf (2Pz + zPj) +J(PsY &1 , . 



" 



for a general solution of the proposed problem. (The parame- 

 ter a is involved in P,z.) 



