202 Mr. I. Todhunter on a Problem in 



will be found that by changing the semicircle into a semiellipse, 

 a solid can be formed of greater volume with an equal surface. 



In the first example which I brought forward, I supposed the 

 length of the axis of the required solid of revolution to be indefi- 

 nitely small, and I showed that the result which 1 am examining 

 is incorrect. If the length of the axis be finite, it may be shown 

 that, by changing the segment of a circle into a segment of an 

 ellipse, a solid can in many cases be formed greater than that 

 which is erroneously said to be the maximum, but having an 

 equal surface. I do not say that this is possible in all cases ; 

 but the fact that it is possible in any case establishes the opinion 

 which I am maintaining. 



Of course I do not assert that in the cases I have noticed 

 I have here assigned the greatest solid, or a maximum solid ; I 

 have only professed to show that the statements on which I am 

 commenting are erroneous. 



The foregoing examples will enable a person who has not 

 studied the Calculus of Variations, but is acquainted with the 

 elements of the Differential and Integral Calculus, to form an 

 opinion on the subject I am discussing. I shall proceed, in the 

 second place, to show that the method by which the erroneous 

 results are obtained is essentially unsound. 



Let 



u= \vdx, 



where v is a function of x, y, and the differential coefficients of y 

 with respect to x. Then by the Calculus of Variations we obtain 



Bu = j A (8z/ —pSx) dx + B, 



where B stands for certain terms which are free from the inte- 

 gral sign, and A is a function of %, y } and the differential 

 coefficients. 



Now if u is to be a maximum or minimum, we must have, ac- 

 cording to the received theory, A = 0; and this is admitted in 

 the article in the July Number. And if there are more than one 

 value of the ordinate corresponding to a given abscissa, the rela- 

 tion A=0 must in general be satisfied at each point thus assigned. 



Suppose that y' and y" represent two values of y which cor- 

 respond to one abscissa x ; and let A' and A" denote the corre- 

 sponding values of A. Then we must have A' = and A" = 0. 

 This is also admitted in the article in the July Number; the 

 equations A'=0 and A" = are expressed at full on page 52. 

 Then from these the equation A'— A" = is deduced, and the 

 following words are added : — " This last equation, inasmuch as 

 it takes account of both values of A, is the one which the form 

 of the curve is required to satisfy. To draw any inference from 



