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



Let P be the point of projection, PG the normal to the surface 

 at P, PN = y the ordinate of P at right angles to the axis of 

 revolution, Q the point where the osculating plane of the path 

 meets the axis. Let Z GPN = a, and Z GPQ <f>. 



When the particle is projected along the tangent to the 

 circular section with velocity F there is initially no acceleration 

 along a line in the meridian plane at right angles to PQ. 



Hence resolving along this line we have 



R sin <f> mg cos (a <) = 0, 



where m is the mass of the particle, and R is the pressure. 

 Again, resolving along PN, we have 



F 2 

 m cos (a <f>) = R cos a, 



where p is the radius of curvature of the path. 

 Now, with the notation of Article 205, 



Also y = PN=PGcosa. 



Hence tan </> = gy\ F 2 . 



This equation determines the position of the osculating plane 

 of the path. 



Now if tan < > tan a, or F 2 < gy cot a, the osculating plane of 

 the path initially lies below the horizontal plane through the 

 point of projection, and if tan </> < tana, or F 2 >#i/cota, it lies 

 above that plane. 



*207. Examples. 



1. A particle moving on a surface (smooth or rough) under no forces but 

 the reaction of the surface describes a geodesic. 



2. A particle moves on a rough cylinder of radius a under no forces but 

 the reaction of the surface, starting with velocity V in- a direction making an 

 angle a with the generators ; prove that in time t it moves over an arc 



a/z" 1 cosec 2 a log (1 +/i Vta~ l sin 2 a), 

 fi being the coefficient of friction. 



3. A hollow circular cylinder of radius a is rough on the inside, and is 

 made to rotate uniformly with angular velocity o> about its axis which makes 



