52 KINETICS OF A PARTICLE. [95. 



equation independently of the time, it follows that the motion is 

 necessarily plane whenever the conditions (22) are satisfied. The 

 constants of integration c v c 2 , c 3 are evidently proportional to 

 the direction-cosines of the normal to the plane of motion. 



95. If the equations (23) be written in the form 



\ydz- zdy _ f \ zdxxdz _ , \xdy-ydx_ , 

 2 dt 1 ' 2 dt 2 ' 2 dt 



they show that the projections of the motion on the three 

 co-ordinate planes have constant sectorial velocities <:/, c%, c s ' ; 

 hence, the sectorial velocity of the motion itself is constant, viz. 



dS / 



It follows in this case that the sector SS Q , described during 

 the time tt^ is proportional to this time : 



5-5 =^' 2 +f 2 ' 2 +V 2 ('-'o)- (25) 



96. 'Exercises. 



(1) A particle of mass m is attracted, according to Newton's law, by 

 a mass m' concentrated at a fixed point O. If x , y , Z Q be the initial 

 co-ordinates, and x , y , Z Q the initial velocities of the particle, find the 

 equation of the plane in which it moves, and show that this plane passes 

 through O and the initial velocity. 



(2) A particle is attracted by n fixed centres, whose forces are directly 

 proportional to the masses of the centres and to the distances from 

 them. Show that there is one position of equilibrium for the particle, 

 and that the motion takes place as if the total mass of all the centres 

 were concentrated at this point. Find also the equation of the plane of 

 the motion. 



(3) A particle is acted upon by a central force, i.e. by a force whose 

 direction passes through a fixed point, and whose magnitude is a func- 

 tion of the distance from this point, say F=mf(r). Show that the 

 path is a plane curve, and find the equation of the plane of the motion. 



(4) The equation (15) can, by Art. 89, be written d(mvp) /dt=xYyX. 

 Show that the two terms of d(mvp}/dt mpdv/dt + mvdp/dt are equal 



f 



