98 KINETICS OF A PARTICLE. [181. 



181. In the most general case, the given forces X y Y, Z will 

 depend not only on the position of the particle, but also on its 

 velocity and on the time. In this case, Q would be a function 

 of q, dq/dt, and t\ and equation (17) represents a differential 

 equation of the second order between q and /. 



If, however, the resultant Fof. the given forces depends only 

 on the position of the particle so that Q is a function of q 

 alone, the right-hand member of (17) is an exact differential, 

 and a first integration can at once be performed. Then, substi- 

 tuting for ?/ 2 its value from (14) in terms of q and dq/dt, we find 

 a differential equation of the first order whose integration gives 

 / in function of q. 



182. Exercise. 



A particle of mass m is constrained to a common helix x = a cos 0, 

 y = a sin 0, z = *0, whose axis is vertical. The particle is subject to 

 gravity and is attracted by a centre situated on the axis, with a force 

 directly proportional to the distance. Determine the motion. 



3. MOTION ON A FIXED SURFACE. 



183. Just as for motion on a curve (Art. 172), we find the 

 general equations of motion 



dP ' r ds* 



The normal reaction 



N=^/N*+N?+N* (2) 



being at right angles to the constraining surface 



