238 CENTRAL FORCES; 



It may be reduced to five kinds of formulae. 



1. If the central force in two similar orbits be called F and 

 f, the times T and t, the versed sines of half the arcs S and s, 



O Q 



then F :/::: ; and generally F is as ^. 



2. But draw S P to any given point of the orbit in the 

 middle of an infinitely small arc Q C. Let T P touch the 

 curve in P, draw the perpendicular SY from the centre of 

 forces S to P T produced, draw S Q infinitely near S P, and 

 Q E parallel to S P, Q o and E o parallel to the co-ordinates 

 S M, III P. Then P being the middle of the arc, twice the 

 triangle S P Q is proportional to the time in which C Q is 

 described. Therefore QPx SYorQL xPSis proportional 



C 

 to the time ; and Q E is the versed sine of , therefore F as 



2i 



S O E 

 7 becomes F as -=^- ^-^ ; and if S M = x, M P = y, and 



-*-' y X O -L 



because the similar triangles QEo and SMP give QE = 



x 



, and because A M being the first differential of S M, 

 lifferential (negatively), therefore Q E = 

 (taken with reference to dt constant), and 



is as But L 



x x LQ 2 x ( 



L P is the differential of S P or V^ 2 + f. Therefore L Q 2 = 



y 



But as the differential of the time (L Q x P S) may be 

 made constant, Q E will represent the centripetal force ; and 



