Principles of the Calculus of Variations. 51 



this equation carries the other with it. I proceed now to the 

 investigation of the line which gives the absolute maximum. ,. : 



For this purpose we must use the integral of the equation 

 A = 0, or, changing for convenience the sign of X, the equation 



2ydx = \dx\/l+p 2 — d. 7 J^-= - 



By integrating, we obtain the indefinite integral 



J a a/1 +p 2 



It may now be assumed that the line has a limit in the di- 

 rection of the axis of x, and that the abscissa of the extreme 

 point is determined by making p infinite. Let y be the corre- 

 sponding ordinate, and let s and the area \ydx commence at that 

 point. Then the arbitrary constant C is equal to \y . The 

 other limit of the integration may, if we please, be at the point 

 for which s = s ] andjo=0; in which case we obtain the definite 

 integral 



2§y' dx=\s l +\y . 



The supposition of a single value y of the ordinate at the 

 limit for which p is infinite, implies that for other values of the 

 abscissa there are two values of the ordinate. The above inte- 

 gration has been performed with reference to those values of the 



ds 



ordinate for which -r-, or v/l -t-p" 2 , and p are positive, which set 



of values I have designated by the symbol y ! . But when the 



integration has reference to the other set of ordinates, which I 



ds 10 



shall call y", -j- and p will change sign, while will retain 



ax v 1 -\-pr 



the same sign. Hence the integral taken from 5=0 to 5= — s 2 , 



and from p= infinity to p = 0, gives a second definite integral, 



viz. 



2\y n dx = — \s% + \y . 



By subtracting this from the other, the result is 



f:(y-jr")<**= !&+**)• 



Now it is evident that this equation is satisfied by the suppo- 

 sition that the line is a circle. For in that case, since ^ = for 

 the extreme ordinates, the area is a semicircle, s } +s 2 is the length 

 of the semicircular arc, and X is the radius of the circle. If the 

 integrations be taken from the ordinate y to any ordinates y x 



E 2 



