254 



CENTEAL FORCES; 



their places at given times. This branch of the subject, 

 therefore, divides itself into two parts; the one relating to 

 motion in the conic sections, the other to the motion of bodies 

 ascending or descending under the influence of gravitation. 



In order to find the place of a revolving body in its 

 trajectory at any given time, we have to find a point such 

 that the area cut off by the radius vector to that point shall 

 be of a given amount; for that area is proportional to the 

 time. Thus, suppose the body moves in a parabola, and that 

 its radius vector completes in any time a certain space, say in 

 half a year moves through a space making an area equal to 

 the square of D ; in order to ascertain its position in any 

 given day of that half year, we have to cut off, by a line 

 drawn from, the centre of forces, an area which shall bear to 

 D 2 the same proportion that the given time bears to the half 

 year, say 3 to m 2 , or we have to cut off a section A S P = 



g 



D 2 , A P being the parabola and S the focus. This will be 



m 



done if A B be taken equal to three times A S, acd B being 



drawn perpendicular to A B, between B 0, B A asymptotes, a 

 rectangular hyperbola is drawn, HP, whose semi-axis or 

 semi-parameter is to D in the proportion of 6 to m ; it will cut 

 the parabolic trajectory in the point P, required. For calling 

 A M = x and P M = y and A S = a ; then A B = 3 a and y x 

 (# -+- 3 a) = half the square of the hyperbola's semi-axis, 



6D 36 D 2 18 D 2 

 which axis being equal to , y (x + o a) = - = , 



m 2 m 3 mr 



