6 CoLOKADo College Studies. 



along the tangeut is 



— r sill cos tv^. 



Multiplying by maSs, we have 



— r sin cos w^ni. 

 Equating and simplifying, the resulting equation is 



Y i— = r \-2 iiv—r sin cos ^ iv . 



prdo <Jt 



The forces along the <f co-ordinate are 

 i?m= impressed force, 



sin^'?' — m=effective force due to the accelera- 

 dt 



tion of angular velocity iv, 



dp 



-»i=difference in pressure on the op- 

 f>r sin Od<p 

 posite faces. 



Since by the combined action of u and ic a particle falls 



behind , ■ ,, i 



(U- sm Oa<p, 



which, converted into terms of force, gives 



2 sin wu (see note, p. 3), 

 we obtain a total force contributed by the combined action 

 of these two velocities equal to 



2 sin toil m; 

 V also carries the particle through the distance EF, and since 

 the line FK is shorter than EH there will be a resultant 

 effective force due to the combined action of v and iv. 

 EH=r sin d(f, 

 FK=r sin {0— do ) d<p, 



= rd<p (sin cos do— cos o sin do). 

 Substituting, for cos do, 1 and, for sin do, do 

 FK=rd<p sin O—rdf cos do, 

 EH—FK=rd<p cos do, or the distance through 

 which the particle may be said to fall under the combined 

 action of n and v. This is equivalent to a force 



2 r cos ivv, 

 or for the mass of the element, 



2 r cosOwvm. 



