226 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



where M is the total mass of the body. By the construction of 

 57, 59 the resultant of parallel vectors P and Q is applied at the 

 center of mass of masses proportional to P and Q placed at their 

 points of application. Consequently the various elements being pro- 



portional to the masses m, 

 this component of the 

 momentum is applied at 

 the center of mass of the 

 body. 



There remains the 

 component of momentum 

 perpendicular to the instan- 

 taneous axes. Let OZ 

 (Fig. 68) be the instan- 

 taneous axis, and let r be 

 the perpendicular distance 

 from it of any point ^P f 

 and let the angle made 

 by r with the X-axis be #. 

 Now P is moving parallel 



Fig . 68 . to the XY- plane with the 



velocity v = r& perpendic- 

 ular to r, so that the projections of this velocity are 



v x = v sin & = G)r sin # = 



v 



vcos& 



or cos -9- 



ox. 



Thence we obtain the components of momentum 



M x = 2 may = oZmy = May, 



52 ) K/r _ 



where x, y are the coordinates of the center of mass. The resultant 

 momentum is accordingly equal and parallel to the momentum that 

 the body would have if concentrated at the center of mass, but its 

 point of application is different, for the components M x , M y are not 

 applied at the center of mass, inasmuch as their elements are pro- 

 portional, not to m but to my and mx. The magnitude of the 

 resultant momentum being given by M x , M y , M~, we may find its 

 axis by obtaining its three remaining coordinates, representing the 

 angular momentum. We have 



53) H y = 

 H z = 



MVy 



= MVx 



= &2m (x 2 -f- y 2 ) = 



