GENERAL PRINCIPLES 

 J 



X' 



nr 



A-'' 



Calling the angle between the two directions a, we have 



MV' = MV cos a; 

 this angle is said to be a directing cosine of MV. 



More generally, the moving point goes from M to M' following 

 the diagonal of a cube or parallelepiped (fig. 9) ; we want to know 

 its speed in relation to the three di- 

 mensions of the figure, OX, OY, OZ. 

 For this it is necessary to project MM' 

 by two lines parallel to OZY, which 

 will give mm' and by two lines parallel B <-' 

 to ZBC which will give m^m\, and 

 lastly by two lines parallel to BYD, 

 which gives m 2 m' 2 . If, therefore, the 

 speed of the given vector is MM ', we 

 can easily calculate the resolved speed 

 along the three rectangular axes. If 

 we call the directing cosines of the 

 vector MM 1 a, p, y we shall have 

 mm' = MM' cos a, myn\ MM' 

 cos , m t m' 2 = MM' cos y. 



4. Equations and Diagrams. Any movement can be defined 

 by an equation ; thus a simple harmonic motion is : 



s = a . sin 2 TC * 

 A rectilineal movement with uniform acceleration is : 



-- X 



FIG. 9. 



or movement at constant velocity is : 



s = vt. 



Motion can also be shown by a diagram or graph. Let there 

 be two lines, OX and OY, at right angles to one another, the line 

 OX is termed the abscissa and the line OY the ordinate ; they 

 are both called axes of co-ordinates and the point O is their 

 origin (fig. 10). If we plot time as abscissae, that is to say, the 

 values of / from zero to T (which is the period), and the corres- 

 ponding values of the displacement as ordinates, we shall have 

 the curve OT. If t = O, then s = 0. 



