21] MOTION ON SPHERE. 43 



^-l^-lx 

 dt* dx~ ^> 



d'g . 8 <f , 

 -W=*Tz+9 = Kz + 9- 



Now inserting these in 36) we have to determine 



w+f+*)+ 



and since x 2 + y 2 + # 2 = Z 2 , 



Using this value of I in the differential equations 40) we have to 

 integrate 



dt 



42) 



Now differentiating the equation of constraint 39) by t gives 



An\ dx . di/ , ds r. 



43 ) x dt+y^ + g w=- 



Multiplying the equations 42) respectively by ~,~? -^f> ^? adding 

 and making use of 43) we may integrate at once and obtain 



where In is an arbitrary constant of integration. This integral gives 

 us the square of the velocity and shows that it depends only upon 

 the initial velocity and the height through which the particle has 

 fallen, for if it has a velocity V Q when z = # , we have 



to determine h. 



Making use of 44) in 41) we have 



45) Jt- 

 and from 38) 



46) J? = 



