304 On solving some Problems in the Calculus of Variations. 



be immediately determined. With reference to this point, Pro- 

 fessor Cayley says, " after the evolute is obtained, we must take 

 not any involute, but the proper involute of such evolute."" But, 

 as Professor Cayley well knows, there is no proper involute of 

 any evolute, because the number of the involutes of a given evo- 

 lute is unlimited. What he calls " the proper involute " is the 

 curve given by integrating the equation Ap = 0; and on the 

 before-mentioned gratuitous assumption that this equation and 

 A = are identical, he refuses to recognize any involute derivable 

 from the latter equation other than that curve. Of course I do 

 not admit that this is an argument, because, for the reasons 

 already urged, I maintain that there is no ground for the initial 

 assumption. 



The equation of the involute which gives the solution of Pro- 

 blem II. contains three arbitrary constants, because it involves 

 the arbitrary length of the cord which, by unwinding from the 

 evolute, describes that involute. By eliminating the three con- 

 stants a differential equation of the third order is obtained, which 

 is evidently not identical with the equation A = of the second 



dA 

 order. Neither is it identical with -j- = 0, because it cannot be 



satisfied if A = a constant. But it is found that that equation 



d u d u 



of the third order is verified by substituting for -^-| and -^\ the 



values of these differential coefficients deduced from A = 



dk. 

 and its derived equation -^—=0. This is proof that the involute 



which may be regarded as the solution of the problem is strictly 

 derived from, and exclusively depends upon, the equation A = 0. 

 The process of derivation by the intervention of an evolute I have 

 called " a new integration," as being distinct from the mode of 

 solution by ordinary integration. 



The mathematical reasoning with which Professor Cayley con- 

 cludes his communication only amounts to a proof that the in- 

 tegral of Ap = gives a curve which is included among the invo- 

 lutes of the evolute which was derived from the equation A = 0. 

 I have no remark to make on this result, as I had already ob- 

 tained the same in the article in the July Number. 



For the reasons above alleged, I adhere to the statement made 

 at the end of the former communication, namely that I have 

 succeeded in removing from analytics the reproach of failing to 

 solve certain problems in the Calculus of Variations. 



Cambridge, September 6, 1871. 



