318 PBOPOSi ON. 



III. I.' :' .nn of a hoi: U solid nf revolu- 



voliime, which shall exercise tli- >t at- 



traction in a given direction on a L'ivcn p 

 tion varying as any inverse power of the distance. 



Take t'. jiartiel.- a< the origin, and the Lfivcn direc- 



tion as the line from which to mea-ure ai 



distance; let r, 6 be the polar co-ordinates of any point in 

 d plane passing through the ^iven direction. r l'hen if 

 : traction vary inversely as the n* power of the distance, 

 the attraction of an element whose co-urdi nates are r and 6 



may be denoted by m and the resolved part of this attrac- 

 tion in the given direction will be ~ cos 6. Hence the 

 equation 



^ cos 6 = constant 



represents a curve such that a given element placed a- 

 point of it will exert the same attraction on the pv.-n particle 

 along the given direction. Hence this equation will n-p 

 the curve which by revolving round the given din 'tion will 

 generate the required solid of greatest attraction, the con-taut 

 determined so as to give to the solid the prescribed 

 volume. It is obvious that such is the case, because the 

 surface we thus obtain separates space into two parts, and 

 any element outside the surface exercises a l<-ss attraction 

 along the given direction than it would if placed within 

 the surface. 



Som references connected with this problem will l.e found 

 in the History of the... Calculus of r&rmioiu page 485. 



IV. Every element of the arc of a polar curve attracts 

 with a force which varii-s inversely as tne n Ul power of the 

 distance: determine the form of the curve when the resultant 

 attraction of any arc on a particle at the pok bisects the 

 between the raaii vectores of the extremities of the arc. 



Take the pole as origin, and any straight line thro 

 the initial line. Let r and be the polar co-ordina: 



