4 - I. KINEMATICS OF A POINT. LAWS OF MOTION. 



a very small interval of time in a new position at a finite distance 

 from the old. The functions are also supposed to have definite 

 derivatives for every value of t. 



Since the motion of a point involves four variables, Kinematics 

 was called by Lagrange Geometry of four dimensions. We shall 

 not here discuss the nature of time, nor the mode of measuring it, 

 reserving the latter until we have considered motions that actually 

 occur in nature, upon which all methods for measuring time are based. 



We may accept the fact that the idea of time, like that of 

 space, is the intuitive possession of us all. Its exact definition must 

 depend on the science of dynamics. 



3. Scalars and Vectors. In mathematics we have to consider 

 two sorts of quantities, those which do not involve the idea of 

 direction, called by Hamilton scalars (because they may be specified 

 by numbers marked off on a scale), and those which do, called steps 

 or vectors. The distance between two points x ly y 1} # 1; # 2 , y. 2 , z. 2 



is a scalar, whereas the geometrical difference in position of the two 

 points is known only when we specify not merely the length, but 

 also the direction of the line joining them. This is usually done by 

 giving its length s and the cosines of the angles made by the line 

 with the three rectangular axes, 



cosA, cos/i, costs,* 

 which in virtue of the relation 



3) cos 2 ;, + cos 2 /i -f cos 2 ^ = 1, 



leaves three independent data. We may otherwise make the speci- 

 fication by giving the three projections of the line upon the co- 

 ordinate axes, = s cos yl = # x 



Squaring and adding we have in virtue of relation 3) 



By the vector AB we mean the line in the direction from A 

 to By and its projections have the sign of the coordinates of B 

 minus those of A, the vector being defined as that which carries us 

 from A to B. We may write symbolically 



pt- A -f AB=pt-B 



AB=pt'Bpt-A. 



AB is to be understood, vector AB. Similarly when we wish to 

 specify that s is to be regarded as a vector (i. e. its direction is to 

 be considered as well as length), we shall write "s. 



