200 Mr. I. Todhunter on a Problem in 



blem as thus enunciated ; and I think it will be admitted that 

 this solution does give the solid of greatest volume, and is in 

 harmony with the recognized principles of the Calculus of 

 Variations. 



In the Philosophical Magazine for July, a condition is attached 

 to the enunciation, as will be seen from the following words 

 which occur towards the beginning of the article: — "That a 

 solid exists, the largest of all solids of revolution whose surfaces 

 are of given area and extend continuously from one extremity of 

 the axis to the other, there can, I think, be no reason to 



doubt " The result which is obtained is that the required 



solid is that which is generated by the revolution of a segment 

 of a circle round its chord, the chord being the straight line 

 which joins the two fixed points. This result is called the abso- 

 lute maximum. 



My present design is briefly to test the accuracy of this result, 

 and the method by which it is obtained. 



I may remark that the word absolute does not seem very ap- 

 propriate, because it naturally suggests freedom from any restric- 

 tion, whereas the result is only maintained with the restriction 

 that the surfaces considered shall be continuous; but this is not 

 a matter of great importance in connexion with my design. 



I will first show, by examining a particular case, that the 

 asserted result is erroneous. 



The particular case I take is that in which the distance be- 

 tween the two fixed points is indefinitely small, so that, in other 

 words, the generating curve is only required to meet the axis at 

 one point. Then the assertion is that the solid of greatest 

 volume with a given surface is that formed by the revolution of 

 a circle round a tangent; on the contrary, I maintain that by 

 taking an ellipse of very small excentricity, a solid can be formed 

 of greater volume with an equal surface. 



Let 2a and 2b be the axes of the ellipse, and let it revolve 

 round the tangent at one end of the axis minor. 



Then the volume generated is 



2nrb x irab, 

 and the surface is 



27Tb x 4a I s/ (1 - e 2 cos 2 <f>) d<f>, 



i 



where e is the excentricity. 



Let r be the radius of a circle; then the corresponding volume 

 and surface are 27r 2 r 3 and 47rV 2 respectively. 



]Now I shall show that if the volumes of the two solids arc 



