205, 206] 



MOTION ON A SURFACE. 



193 



tangent to the path the resolved part of the acceleration along the 

 normal to the surface is the resolved part in that direction of the 

 acceleration along the principal normal to the path, it is there 

 fore 



v 2 



- COS (f). 



Also by a well-known theorem we have p = p cos $. 



Hence the acceleration along the normal to the surface is v*/p , 

 and the pressure is determined by resolving along the normal. 



*206. Osculating plane of path. In Example 1 of 

 Article 204 it is stated that a particle may be projected along a 

 horizontal tangent of a smooth surface of revolution whose axis is 

 vertical with such velocity that it describes the circular section 

 under the action of gravity and the reaction of the surface. It is 

 almost obvious that if the velocity exceeds that requisite for 

 description of the circle the path of the particle rises above the 

 circle, otherwise it falls below the circle. We may use the result 

 of Article 205 to find the position of the osculating plane of the 

 path for any velocity of projection. 



Fig. 53. 



L. 



13 



