Principles of the Calculus of Variations. 53 



the same equation is obtained as that which, it has already been 

 shown, indicates that the curve is a circle. In the case of the 

 function Ap, there were not two equations corresponding to two 

 sets of values, because by the condition of symmetry the two or- 

 dinates to the same abscissa were equal with opposite signs, and 

 the same form of the function consequently included both sets of 

 values. 



When I made the communication contained in the March 

 Number, I had not recognized the distinction which I have now 

 drawn between the conditional and the absolute maximum, and 

 in consequence supposed the first part of the solution to apply 

 to the case in which the given surface h 2 is less than 47rc 2 , the 

 surface of the sphere whose diameter =PQ, and the latter part 

 to apply when h 2 is greater than 47rc 2 . But the present reason- 

 ing has shown that the determination of the absolute maximum 

 is not subject to restriction, and that the line which generates 

 the surface of the maximum solid is a segment of a circle of 

 which the line joining the given points is the chord, whether h 2 

 be greater or less than 4-7rc 2 . The other conclusions arrived at 

 in the former paper, especially the interesting one respecting the 

 form of the maximum ring of given superficies, I have not seen 

 reason to retract or modify. 



It has already been stated in the March Number that the 

 maximum solid obtained in the first part of this investigation, 

 namely, the cylinder with hemispherical ends, is found by ana- 

 lytical calculation to be greater than a solid of the form of a 

 prolate spheroid, whether the eccentricity be very small or nearly 

 equal to unity. I have since ascertained that it is also greater 

 than the solid which has for the generating line of its surface a 

 circular arc connected with the given points by two tangents at 

 its extremities of equal length. If, however, the generating line 

 be a circular arc at right angles to the axis at one of the given 

 points, and connected with the other by a tangent at its extre- 

 mity, the solid will be found to be greater than the cylindrical 

 one. For, in fact, this line, though continuous, does not, like 

 the preceding one, satisfy the condition of symmetry with respect 

 to a perpendicular axis. Moreover, I have ascertained that this 

 unsymmetrical solid is less than that the generating line of 

 whose surface is the segment of a circle passing through the 

 given points ; which, according to the foregoing solution, is the 

 absolute maximum of the solids of continuous form. 



It may be further remarked that as a sphere is the greatest 

 solid of given superficies, there may be all sorts of solids of revo- 

 lution of discontinuous form intermediate in magnitude between 

 the sphere and the solid which I call the absolute maximum. 

 For instance, let the generating line be a semicircle connected 



