90.] PRINCIPLE OF AREAS. 49 



turn about any fixed point is equal to the moment of the resultant 

 force about the same point. 



90. The most important case in which the integration in (16) 

 can be performed is the case when 



xY-yX=o, (17) 



which evidently means that the direction of the resultant force 

 F passes through the origin. If this condition be fulfilled, 

 equation (16) reduces to the form 



dy dx 



mx dt~ my Tt= c ' 



where c is a constant of integration to be determined from the 

 initial position and velocity. 



Kinematically, this equation means that the sectorial velocity 

 remains constant. It can be put into the form 



dS 



dt 2m 

 whence, by integration, we find 



S-S^(t-t^. (19) 



Hence, if the acceleration passes constantly through a fixed 

 point, the sector S S described about this point in any time t 1 

 is proportional to this time. 



This is the principle of the conservation of area for plane 

 motion. 



Dynamically, equation (18) means that if the resultant force 

 passes constantly through a fixed point, the angular momentum 

 about this point remains constant. The proposition can also be 

 called the principle of the conservation of angular momentum. 



If VQ be the initial velocity, / the perpendicular to V Q from 

 the fixed point, equation (18) can be written in the form 



PART III 4 



