Calculus of Variations by a New Method. 223 



In the first place, it may be asserted, in consequence of the 

 previous reasoning, that the equation 



2ydx + Xda*/T+]fi-d. Jf^ =° 



must not now be integrated by means of a factor. It does not 

 appear to admit of exact integration by any other mode of treat- 

 ment; but obviously it is allowable to integrate it under its 

 present form, and thus obtain an expression for the area of the 

 curve. After changing, for convenience, the sign of the arbi- 

 trary constant X, the integration gives 



^=^-VYT? +c > • • • • < A ) 



s being the length of an arc of the curve reckoned from an arbi- 

 trary origin. It is next required to discover from this equation 

 the nature of the curve to which the integral \ ydx belongs. The 

 following reasoning employed for this purpose is of a novel cha- 

 racter, but I think that, upon consideration, every step of it will 

 be found to be both legitimate and necessary. First, it may be 

 remarked that the kind of surface which can alone satisfy the 

 conditions of the problem must be such that its generating 

 curve has two ordinates to the same abscissa. This will be at 

 once apparent by conceiving the distance between the given 

 points to be very small, and the given surface to be large. Next, 

 as the equation (A) cannot apply to one of the ordinates rather 

 than the other, it must embrace both. The necessity for a con- 

 sideration of this kind is recognized by M. Lindelof in his 

 Lec.ons de Calcul de Variations, p. 224. Again, as the right- 

 hand side of the equation contains the arc s } the integration 

 necessarily proceeds continuously from one extremity of the arc 

 to the. other. Now all these conditions are fulfilled by integra- 

 ting relatively to the ordinates of successive points of the curve 

 from one limiting ordinate to another having the same abscissa, 

 the common abscissa being any that we please. Accordingly 

 the result of the integration may be put under this form, 



the right-hand side of the equation being a function of the arbi- 

 trary abscissa. But from the theory of the quadrature of curves 

 it is known that by this operation, regard being had to the signs 

 of the areas, we obtain a segmental area cut off by a straight line 

 coincident in direction with the limiting ordinates. The ques- 

 tion now reduces itself to finding a curve the segmental area of 



