126 KINEMATICS. [229. 



The proof of the converse, viz. that if in a plane motion 

 the areas swept out by the radius vector drawn from a fixed 

 point be proportional to the time, the acceleration must con- 

 stantly pass through that fixed point, is left to the student. 



229. It is well known that Kepler had found by a careful 

 examination of the observations available to him that the orbits 

 described by the planets are plane curves, and the sector described 

 by the radius vector drawn from the sun to any planet is propor- 

 tional to tJie time in which it is described. This constitutes 

 Kepler's first law of planetary motion. 



He concluded from it that the acceleration must constantly 

 pass through the sun. 



230. To express the value of the constant of integration h in ; 

 terms of the given initial conditions (Art. 224), i.e. by means 

 of r , v, i/r Q , we notice that, at any time /, 



dO rdQ ds / ^ 



' v ' (54) 



hence for the time /=o, we find 



^ = Vo sin ^o- (55) 



Denoting the perpendiculars let fall from O on v and v by ; 

 / , /, we have r sin^ =/ , rsin-v/r=/; hence also 



k=PM=pv, (56) 



i.e. the velocity at any time is inversely proportional to its distance \ 

 from the centre. 



231. The equations of motions (49), if multiplied by dx/dt, 

 dy/dt and added, give an equation in which both members are 

 exact derivatives. On the left we find 



