108 Prof. Challis on the Solution of a Problem 



and the remainder had crumbled into sand and has been thrown 

 away. When it fell it broke into three pieces, and was cold and 

 saturated with wet; it was seen to fall by a ploughman of Dr. 

 M'Clintock's, and immediately afterwards picked up by him. 



R. P. Greg. 



Manchester, July 12, 1861. 



XVI. Solution of a Problem in the Calculus of Variations. 

 By Professor Challis*. 



IN the July Number of this Magazine the Astronomer Royal 

 has called attention to a problem in the calculus of varia- 

 tions, the solution of which presents some difficulty. The method 

 of solution I am about to propose, which appears to meet the dif- 

 ficulty, is, I believe, new. 



The following is the enunciation of the problem as given in 

 Mr. Todhunter's ' History of the Calculus of Variations': — To 

 construct upon a given base AB a curve, such that the superficial 

 area of the surface generated by its revolution round AB may be 

 given, and that its solid content may be a maximum. By the 

 rules of the calculus of variations, the ordinary notation being 

 adopted, the solution of the problem is given by the equation 



S.J {y* + 2ay */T+f)dx=0. 



Integrating from x = to x — x v and equating separately to zero 

 the parts outside and those under the sign of integration, we 

 have 



dx */\+p* 



The first equation is evidently satisfied by the hypothesis that 

 y Sy =0 and y x 8y x = 0. The integration of the other gives 



2ay 



vW =b ~ A o 



b being the arbitrary constant introduced by the integration. 



The next step usually taken in solving this problem is to put 

 & = 0, because at the fixed points A and B y vanishes. This 

 appears to have been done previous to the second integration 

 solely because the equation (1) is not integrable unless b = 0. I 



* Communicated bv the Author. 



