100 KINETICS OF A PARTICLE. [185. 



As the resistance N does no work, the principle of kinetic 

 energy gives 



or, dividing by \ m, 



V* = 2(gZ + h). (7) 



To determine the constant of integration h y we have v v^ when 

 zz^ ; hence 



zs). (8) 



185. Another first integral is found by applying the principle 

 of areas which holds for the projection of the motion on the 

 horizontal ^j/-plane. This appears by considering that N is 

 always directed along the radius of the sphere so that the 

 resultant of N and the weight mg of the particle always inter 

 sects the axis of z (see Art. 93). We have therefore 



where \c is the sectorial velocity of the projection OP' of the 

 radius OP=ron the .rj/-plane. 



186. For the further treatment of the problem it is best to introduce 

 polar co-ordinates (Fig. 26). Let & be the angle between r and the 

 axis of z, <f> that between the projection OP of r on the xy-p\a.ne anc 

 the axis of x ; then 



x= rsin6cos<f>, y = r sin sin <J>, z 

 and x=rcosOcos <f>>0 r sin sin <(><)>, 



y=r cos sin <f>>0 + r sin cos <<, 



Hence ^ = j* +/ + ? = r \fr + sin 2 0.< 2 ), 



xyyx=r* sin 2 &-<j>. 



The first integrals (7) and (9) thus become in polar co-ordinates 



(10) 

 (uj; 



