the Calculus of Variations, 201 



equal, the surface of the former solid is less than the surface of 

 the latter ; this is equivalent to the statement I have made above. 

 We have, by equating the volumes, 



ab L2 = r 3 , 

 that is, 



a 3 (l-e*)=r 3 ; 



thus 



= r(l-e 2 )-i 

 Hence the surface formed by the revolving ellipse is 



n 



87rr 2 (l-e 2 )-^f 2 >v/(l-e 2 cos 2 <£)#; 

 Jo 



and we have to show that this is less than 47r 2 r 2 ; that is, we have 

 to show that 



1 t~2 7T 



(1 — e 2 ) ~t 1 V(l — e 2 cos 2 6) d<f> is less than — , 

 Jo * 



the excentricity e being supposed very small. " 



This may be shown in more than one way : the followiDg will 

 be sufficient. If e is very small, so that we may reject e 4 and 

 higher powers of e } the left-hand expression becomes 



This example is a special case of a general proposition which 

 was enunciated in the Philosophical Magazine for March, and to 

 which attention was again invited in July. The general propo- 

 sition is the following : a ring having a circular transverse sec- 

 tion is larger than any other ring having the same super- 

 ficies, and the same radius either interior or exterior, but a dif- 

 ferent form of transverse section. Now I have just taken a spe- 

 cial case of this general proposition, and shown that in this case 

 the result is erroneous. 



Moreover the general proposition itself is erroneous. For it 

 may be shown in nearly the same manner that by taking, instead 

 of the circle, an ellipse of very small excentricity with its major 

 axis parallel to the axis of revolution, a solid can be formed of 

 greater volume with an equal surface when the interior radius of 

 the ring is given. 



Again, the result which immediately follows in the March 

 Number is also erroneous. A figure formed of a rectangle and 

 a semicircle generates a solid which, under certain circumstances, 

 is asserted to have the greatest volume with a given surface. It 



