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XLIX. On the Figure of the Bullet which experiences the least re- 

 sistance from the Air. By Francis A. Tarleton, Fellow and 

 Tutor j Trinity College, Dublin * P 



TO find a surface of revolution whose volume and the dia- 

 meter of whose base are given, and such that it may ex- 

 perience the least possible resistance in passing through the air 

 in a direction parallel to its axis. The resistance of the air on 

 any element of the surface is supposed to act in the normal 

 to the element, and to vary as the square of the normal velocity. 

 Take the axis of revolution for axis of x, and let 2y x be the 

 diameter of the base, then 



Let 



a 





dx 



— y [ = a minimum. 



and 



f{KI) 



7 dx~\ , . . 



4- by t- ?■«'/ = a minimum, 



Take the origin where the curve meets the axis of x; then x — x l 

 is wholly unrestrained. The Calculus of Variations gives 



(v-^)^-f|{v-^>^o. 



Hence 



but 





.-. c=o. 



Rejecting the factor y = 0, we have the equation 



2 



for the generating curve. Hence y — -. sin 0cos 3 0, if 6 be used 



* Communicated by the Author. 

 Phil Mag. S. 4. Vol. 34 No. 231. Nov. 1867. 2 C 



