192 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



*205. Motion on a surface in general. Suppose that a 

 particle moves on a fixed surface under the action of given forces 

 and the reaction of the surface. 



We may imagine the surface to be covered with a network of 

 curves belonging to distinct families, in such a way that at each point 

 of the surface one curve of one family meets one curve of the other 

 family, and we may suppose the curves that meet in any point to 

 cut at right angles. At any point we may resolve the forces of 

 the field into components along the tangents to the curves that 

 meet in that point, and along the normal to the surface. We may 

 resolve the acceleration along the same lines. 



When the surface is smooth the reaction is simply a pressure 

 along the normal. For a particle moving on a smooth surface in 

 a conservative field there will be an energy equation expressing 

 the velocity in terms of the position. We shall see presently that 

 the pressure is determinate as soon as the velocity is known. 



When the surface is rough there will be two components of 

 friction in the directions of the tangents to the two curves that 

 meet at any point, and the resultant friction has the same direction 

 as the velocity but the opposite sense. Also the resultant friction 

 is equal in magnitude to the product of the coefficient of friction 

 and the pressure. 



We have thus the means of writing down equations of motion 

 of the particle, but the process can in general be simplified by 

 using methods of Kinematics and Analytical Dynamics which are 

 beyond the scope of the present work. We shall therefore confine 

 ourselves to the simplest cases. 



We proceed to investigate a general expression for the resolved 

 part of the acceleration along the normal to the surface. 



Let v be the velocity of the particle, p the radius of curvature 

 of its path. The tangent to the path touches the surface, and we 

 suppose a normal section of the surface drawn through it. This 

 section is not, in general, the osculating plane of the path ; we 

 suppose that it makes an angle &amp;lt;f&amp;gt; with this osculating plane. We 

 take p to be the radius of curvature of the normal section of the 

 surface through the tangent to the path. 



Since the normal to the surface is at right angles to the 



