222 Prof. Challis on the Solution of a Problem in the 



Supposing the integral to be taken from one extremity of the 

 straight line to the other, y-yly^ and y &y are each indetermi- 

 nate, and the equation is satisfied because p = and jt? =0. If 

 taken between either of the given points and the extremity of 

 the corresponding circular arc, and if y Q and y v be the respective 

 ordinates at these limits, the second term of the above equation 

 vanishes because 2/ oV =0 ; but since y^y x is indeterminate, the 

 other term can vanish only because ^ = 0. We may hence infer 

 that the straight line and the arcs are continuous at the points of 

 junction, and that each arc is a quadrant of a circle. The gene- 

 rating line of the required surface of revolution is thus com- 

 pletely defined. 



The result to which the foregoing investigation has conducted 

 may now be stated in these terms : — The solid consisting of a 

 cylinder and two hemispherical ends of the same radius, is larger 

 than any other solid of revolution having the same amount of sur- 

 face and the same length of axis. 



If r be the common radius of the cylinder and the hemi- 

 spheres, and 2c the given distance between the points, the sur- 

 face of the solid is 47rcr, and its content is 27tt 2 ( c— ^ j. Hence 



h 2 

 if h 2 be the given surface, the radius r is equal to - — , and the 



content of the maximum solid of revolution is 



SttcV \2ttc 2 )' 



It may be stated, as confirming the truth of the theorem, that 

 on comparing the volume of a prolate spheroid of small eccen- 

 tricity with that of a solid of the form above determined, having 

 the same superficial area and the same axis, the latter was found 

 to be the greater. A like result was obtained by comparison 

 with a prolate spheroid of eccentricity nearly equal to unity. 

 Supposing k to be the ratio of h 2 to Anrc 2 , the calculation for the 

 case of small eccentricity gave the result that the prolate sphe- 

 roid is less than the other solid if (1— k) 3 be positive. Hence 

 k must be less than unity, or h 2 less than Amc 2 . In the limit- 

 ing case, h 2 = 47rc 2 , the solid is plainly a sphere, the cylindrical 

 part vanishing. If h 2 be greater than 47rc 2 , there is nothing 

 corresponding to the cylindrical part, and the foregoing inves- 

 tigation is no longer applicable. But since, for the reasons 

 adduced at the beginning of this communication, there must 

 still be a maximum solid, it is necessary, in order to complete 

 the solution of the problem, to enter upon a separate investiga- 

 tion for this case, which I now proceed to do. 



