i8o.] MOTION ON A FIXED CURVE. 97 



(9) Investigate the motion of a particle subject to gravity, and com- 

 pelled to move on a circle whose plane is inclined to the horizon at an 

 angle a. 



(10) A particle constrained to a straight line is attracted to a fixed 

 centre outside this line, the attraction being proportional to the distance 

 from the centre. Determine its motion. 



180. Motion on Any Fixed Curve without Friction. The posi- 

 tion of a point on a curve can always be determined by a single 

 variable. Thus, for instance, the length s of the curve counted 

 from some origin on the curve might be taken as this variable ; 

 if the curve be a circle, the polar angle 6 might be selected ; 

 on an ellipse, tb^ eccentric angle </> ; on a cycloid, the angle 

 through which the generating circle has rolled, etc. We shall 

 designate this variable by q, and write the equations of the 

 curve in the form 



*=/i(?). y=f*(q\ *=/ 8 (?)- (13) 



The expression for the velocity v is in this case 



/^y 



UJ 



If there is no friction, the real equation of motion is the 

 equation (i) of Art. 169, which is equivalent to the equation of 

 kinetic energy (7), Art. 173 ; when the variable q is introduced, 

 this equation becomes 



where v 2 is given by (14). 

 Putting, for shortness, 



we can write the equation of motion in the simple form 



(17) 



.PART III 7 





