230 



CENTRAL FORCES; 



p b s, as the arcs a 6, bp, which arcs being all as the times, the 

 areas are proportional to those times of describing them, and 

 therefore S and s are the centres of the deflecting forces. 

 Then, drawing the tangents A C, a c, and completing the 

 parallelograms DC, dc, the diagonals of which coincide with 

 the evanescent arcs A B, a b, we have the centripetal forces 

 in A and a, as the versed sines AD, ad. But because A B P 

 and abp are right angles (by the property of the circle), the 

 triangles ADB, APB, and adb, apb, are respectively similar 



to one another. Wherefore AD:AB::AB:AP and A D 



AB* ab* 



= -r-^r ; and in like manner a a - , or, as the evanescent arcs 

 AP ap 



coincide with the chords, A D = arc pr- and a d = arc . 



AP ap 



Now these are the properties of any arcs described in equal 

 times ; and the diameters are in the proportion of the radii ; 

 therefore the centripetal forces are directly as the squares of 

 the arcs, and inversely as the radii. 



It is difficult to imagine a proposition more fruitful in con- 

 sequences than this ; and therefore it has been demonstrated 

 with adequate fulness. 



In the first place, the arcs described being as the velocities, 

 if F, / are the centripetal forces, and V, v the velocities, and 

 R, r the radii, F : / : : V 2 : ; and also : : r : K; or F : / : : 



