the Calculus of Variations. 13 



I submit the following solution, as what I believe to be the 

 real interpretation of the formula? given by the Calculus. It is 

 founded upon these three principles : — 



(1) There is nothing to prevent us from accepting as solution 

 of the problem a discontinuous curve, provided the different parts 

 meet in a way which is suitable to the conditions of the problem. 

 Mr. Todhunter in several places has alluded to such disconti- 

 nuity (see pages 19 and 174). 



(2) If, in the solution given immediately by the Calculus of 

 Variations, we are certain that no accidental or adventitious 

 factor has been introduced ; and if we find that the solution, 



expressed under the form <M x, y, —-, &c. J = 0, is the product 



of two factors, then we are bound to consider each of the curves 

 represented by the two factors as a good and sufficient solution 

 of the problem. 



(3) And, to exhibit the solution in its utmost generality, we 

 must use both the solutions given by these curves, in such com- 

 bination as the circumstances of the problem indicate to be 

 proper. 



I now proceed to apply these principles to the problem 

 before us. 



Since it \dx . y 2 is to be maximum, while 2-7T \dx . y\/{l +P* 2 ) 

 is given, then if a be a constant to be determined hereafter, the 

 value of V will be 



y* + 2ay</(l+p*); 

 and treating this in the usual way, we find 



b-y% 



where b is another constant produced by integration. It is cer- 

 tain here that no factor has been introduced. 



Since the curve is to meet the axis, y=0 at certain points, 

 and \/{I+p Ci ) is never =0. Hence b must =0; and our 

 equation becomes 



2a V _ _ ? .2 



e followi 

 v =0. 



y 



which is satisfied by either of the following, 

 2a 



