226 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



167. A particle moves on a smooth cycloid whose axis is vertical and 

 vertex upwards in a medium whose resistance is (2c) ~ l (velocity) 2 per unit of 

 mass, and the distance of the starting point from the vertex is c ; prove that 

 the time to the cusp is A /{8a(4a c)/^c}, 2a being the length of the axis. 



168. A particle of mass m moves under equal constant forces mf along 

 the tangent and normal to its path, and the resistance is mfv 2 [k 2 when the 

 velocity is v. Prove that the intrinsic equation of the path is 



where u is the velocity of projection. 



169. A particle moves in a medium in which the resistance at any point 

 varies as the density of the medium at the point and as the square of the 

 velocity of the particle, and the particle describes an ellipse under the action 

 of two forces to the foci varying inversely as the nth power of the distance ; 

 find the density of the medium at any point of the path ; and show that if 

 n = l and the forces are equal at equal distances the density varies as the 

 acceleration with which the particle would move if constrained to describe 

 the same ellipse under the same forces but without resistance. 



