206-208] RESISTING MEDIUM. 195 



an angle a with the vertical. Show that a particle can slide down a line 

 parallel to the axis with uniform velocity 



where /z is the coefficient of friction, and /z > cot a. 



4. An ellipsoidal shell whose principal semiaxes are a, b, c (a > b > c) is 

 placed with the greatest axis vertical and a particle is projected from one of 

 the lower umbilics with velocity v along the tangent to the horizontal section 

 within the ellipsoid. Show that the osculating plane of the path is initially 

 above or below this section according as 



v 2 > or < gab 2 (6 2 /c 2 - l)A/{( 2 - c 2 ) (a 2 - 6 2 )}. 



208. Motion in Resisting Medium. In illustration of 

 the class of forces admitted in Rational Mechanics under the name 

 resistances, we consider cases of the motion of a particle in a 

 known field of force when, in addition to the forces of the field, 

 there is exerted on the particle a force proportional to a power of 

 its velocity having the same direction as the velocity and the 

 opposite sense. 



Problems of this kind are related to facts of observation in 

 regard to the motions of bodies in the air and in other fluid media. 

 In many cases it is found that the observed facts can be approxi- 

 mately represented by the supposition that the resistance is 

 proportional to the velocity, this is true for instance for the 

 motion of a pendulum swinging in air. There are other cases for 

 which it is found that the facts are better represented by suppos- 

 ing the resistance to be proportional to the cube of the velocity, 

 this appears in fact to be the simplest function of the velocity 

 which gives rise to an approximate representation of the motion 

 of a shot. 



In such cases there is not necessarily any dissipation of energy 

 in the system. The motion of the body through the fluid generates 

 motion in the fluid, so that kinetic energy is gained by the fluid, 

 and thus the body must part with kinetic energy at a greater rate 

 than it would do in the absence of the fluid. Work is done by the 

 body against the resistance, and an equal amount of work is done 

 upon the elements of fluid, and its equivalent may be produced in 

 kinetic energy of the fluid, potential energy of strain, &c. We 

 can in fact conceive cases in which this would happen. On the 

 other hand in actual cases some of the energy is invariably 

 dissipated, i.e. converted into heat, or generally into other forms of 



132 



