294 Central Forces. 



(N" aN" 



BD = -2~ Xf (25) or because ~^~= const, t is as the area BD, or 



supposing the particle to fall from A, the time from A to B is to the 

 time from C to B as the area of the semicircle ADB to the area BD. 

 In like manner (23) gives his case 2. (see his fig. 2.) the axis AB 

 of his rectangular hyperbola =2a, CB=r, AC=2a-j-^} and 



V 



2a +r ^„^ . P 



cot.(f)^=\/ = tan. ang. CBD, .■.-^ — (p' = CBD, hence as 



r 



2 



before the area BD= "^""^^ (^^^' ^"^^ ^^® ^™® ^"^'^^^ C to B is as 

 the area BD. Also (24) gives his case 3. supposing that p= the 

 semiparameter of his parabola, (see his fig. 3.) r=CB, then 



cot. (p''=\/ -^= tan. ang. CBD .•.-^ — 9'" = CBD, hence as before 



1 have the area BD= -^Xt (27), and the time of motion of the 



particle from C to B is as the area BD. Again, (25), (26), (27), 

 agree with Newton's conclusions, (Prin. Vol. I. Sec. 7, prop. 35,) 



aV"t 

 the areas BD in (25), (26) are equal to ^ ? in which a= the ra- 

 dius of the circle described by the particle at the distance a, from 



latus rectum 

 the centre of force = ^ of the circle, or rectangular hy- 

 perbola, and V^= the arc of the circle described by the particle 



in the time t, .'. — ^ — ==- the area described by the radius vector, a, 



in the circle (rad. =«,) whence his first case is evident; also his se- 

 cond case follows in the same manner from (27), the radius of the 

 circle in this case =p— half the latus rectum of the parabola. Also 

 Newton's constructions of props. 36 and 37, are evident from what 

 has been done, his 37th being equivalent to supposing that V, V, r 

 in (7) and (8) are given to find a ; whence by squaring those equa- 

 tions and writing them in the form of proportions, his proportions 

 will be obtained for finding a, &;c. but as these proportions are very 

 simple, I shall here leave them. 



