230 



less than that on any other frustrum of the same height and 

 base, the motion being in the direction of the axis. 



Produce the frustrum to its vertex, and let 2 be the 

 angle of the cone. Then it is evident that the radius of 

 one end being r, that of the other will be r h tan. 0. And 

 the resistances on the curved and flat ends will be respec 

 tively 



sin. 2 0(r 2 - r- h tan. 0| 2 ) 

 and (r h tan. 0) 2 , 



the common constant factor TT x v 2 being omitted. The 

 sum of these two is easily seen to be 



r^-r h sin. 2 + A 2 sin. 2 0; 

 differentiating this we have 



h sin. cos. = r 2 r sin. 2 0. 

 Put x for the whole height of the cone, then will 



whence this construction. Let O and D be the given 

 centres of the two ends, and C be any point in the circum 

 ference of the base. Bisect O D in Q and produce Q D 

 to S, so that Q S = Q C. S is the vertex of the required 

 cone. This result Newton thought might be of use in 

 the building of ships. 



To find the surface of revolution such that, when alto- 

 gether immersed) the resistance on it ivill be less than on any 

 other surface of the same length and breadth. 



Take the axis of revolution as the axis of or, and let y 

 be the ordinate of the generating curve. And let p be 

 the diff. co. of y with respect to x. Then, omitting con- 



