Calculus of Variations by a New Method, 225 



curve for the case of h 2 being greater than 4ttc 2 had been made 

 by the usual method of integrating a differential equation, it does 

 not appear how the same analysis could have embraced the two 

 cases of h 2 being less and greater than 4*7rc 2 . As it is, the form 

 of the curve for the latter case has been ascertained from an ex- 

 pression for the area as a function of the abscissa, instead of an 

 expression for the ordinate as a function of the same variable. 



From the result of the investigation for the case in which h 2 

 is greater than 4*ttc 2 , it may be concluded that, if the surface of 

 the maximum solid of revolution be required to pass through two 

 points of the axis distant from each other by 2c, the surface is 

 that generated by the segment of a circle, the radius of which 

 may be calculated from the given value 2c of the chord and the 

 given superficial area h 2 of the solid. 



Since the coordinates of the centre of the circle are at disposal, 

 the generating arc may terminate at any two given points not 

 situated on the axis of revolution, and the surface generated by 

 the arc and the coordinates of the points will enclose a maximum 

 solid if the curved part of the surface be given. 



Also, if the two points be equidistant from the axis and be 

 supposed to approach indefinitely near to each other, the gene- 

 rating line becomes a complete circle, and the solid is a ring 

 having a circular transverse section. Hence from the foregoing 

 reasoning it may be concluded that this ring is larger than any 

 other having the same superficies, and the same radius either in- 

 terior or exterior, but a different form of transverse section. 



Let us now suppose that the surface generated by the extreme 

 ordinates and the connecting arc is given, and that the solid 

 enclosed by this composite surface is required to be a maximum. 

 In this case, if r and r' be distances of points of the plane circular 

 areas from the axis, we shall have the additional terms 



to be integrated from r=0 to r = y„ and from i J =0 to r' = y q . 

 Thus the total quantity freed from the sign of integration will be 



and the values of y x and y 2 have to be determined. Hence 

 neither Sy l nor Sy 2 may be equated to zero, and we shall there- 

 fore have 



P± 



+ 1 = 0, _" ^i =0i 



or P%— ~~ & ana< Pi = + °° • This proves that the extreme ordi- 



