10 I. KINEMATICS OF A POINT. LAWS OF MOTION. 



A velocity, like any other vector, may be resolved into com- 

 ponents, 



dx ds dx dx 



x ~ ds dt ds dt 



Vy ~ V &s ~dt ds lit 7 

 dz ds dz dz 



or the projection of the velocity on any direction is the velocity of 

 the projection of the point on that direction. 1 ) We have therefore 



-j r,\ 2 (ds\% (d%\ 2 , (dy\ 2 _\_ jf*V 



' ~ \dt} ~~ \dt) + \dt) " h \dt) ' 



A third method of description of a motion would be to give as 

 before the equations of the path and to give the velocity as a 

 function of the time, 



ds 



An integration of this differential equation would give us 

 s = const +f^(f)dt = <p(t), 



and we should have the same form as before. 

 Fourthly we might have 



18 ^ = 5F = *i(0, <V = ff = -m v> = f; = F s (t) 

 together with the initial conditions 



x = x ^ y = y oj 2=2o } when t = t . 



An integration of these three simultaneous equations would give us 

 a description equivalent to 1). 



In equations 1), if t is any parameter, not necessarily the time, 

 we have what is called the parametric representation of a curve. 

 By the elimination of t, we may obtain two coordinates as functions 

 of the third. If, on the contrary, we have only the path given, 

 whereas the geometry of the motion is known, kinematically the 

 description is incomplete, as the specification of the time is lacking. 

 To remedy this defect of the geometrical representation, Hamilton 

 introduced the Hodograph, which is a curve, the locus of a point 

 related to the moving point on the path by having its position 



1) It is to be noticed that in stating that velocity is a vector we assume 

 the mode of composition of velocities as a matter of definition. 



