40 Integration of Differential Equations of the Second Order. 



have found by a somewhat long ^process, which it would be 



tedious to give in detail here, that by substituting for p and -j- 



dk 

 from the equations A = and -j- = 0, that equation is satisfied. 



It is to be noticed that it is not satisfied if A = a constant ; so 



dk. 

 that this equation of the third order is not identical with — =0. 



dk 

 Since it is necessary to employ both A=0 and -7- =0 for its 



verification the integral from which it was derived may be con- 

 sidered to be a solution of the former equation, and therefore 

 proper for being applied in the present problem. 



The same equation (8) of the evolute would have been obtained 

 if we had employed kp = instead of A = 0. Hence among the 

 involutes obtained must be included the particular one resulting 

 from the integration of the former equation. In fact, the pro- 

 cess may be so conducted as to lead exclusively to that involute. 

 For this purpose it suffices to substitute for 2a — p, in the fore- 

 going expression for p q 3 its value ~^~ -, so as to obtain the 



result 



ky(Py-2af 



1 + / = P2=: 



tf 2 (Py~a) 2 -% 2 (Py-2«) 2 



which only differs in form from the first integral of Ap = 0. 



Just as in the solution of Problem II., the required curve may 

 be described either by unwinding a cord from the evolute given 

 by the equation (8), or by drawing a curve equidistant at all 

 points from that given by the integration of Aj» = 0. The simi- 

 larity of the processes in the two instances is some evidence of 

 the correctness in principle of the new method of integrating. 

 I have not leisure to discuss in more detail the solution of 

 Problem III., and shall only add that when the two given points 

 are on the axis, if the length of the curve be not much greater 

 than the distance between them, the solution shows that its form 

 is like the arc of a bow ; but if much greater, that the form ap- 

 proaches that of a circle. 



I think I may say that I have at length succeeded, after re- 

 peated failures, in removing from analytics the reproach of being 

 incapable of solving this class of problems. Perhaps the pecu- 

 liarity of the requisite process may account for its being so long 

 undiscovered. 



Cambridge, June 17* 18/ It 



