244 CENTRAL FORCES ; 



is equal and opposite to the centripetal. Also the velocity in 



o 



uniform motion, like that in a circle, being as , the space 



c 



divided by the time, and the arc being as the radius r, f is as 



,S 2 T 



5 or as -. If two bodies moving in different circles have 



y* v v 



the same centrifugal force, then the times are as *Jr. It 

 is to the justly celebrated Huygens that we owe the first 

 investigation of centrifugal forces. The above propositions, 

 except the second, are abridged from his treatise.* 



i. First, where the centre of force is the centre of the tra- 

 jectory. In exemplifying the use of the formulas we have 

 shown the proportion of the force to the distance in the conic 

 sections generally, their foci being the centres of forces. Let 

 us now see more in detail what the proportion is for the 

 circle. Let S be the centre of forces and K of the circle, P T 

 a tangent, SY a perpendicular to it, KM and MP co-ordi- 



nates, SK = b, KO = a, PM = y, and MK = x. Then, by 



ST x KP 

 similar triangles, TKP and TSY, we have SY = ^ , 



y* 



or (because the sub-tangent M T = , and a* = x* + */*) 



oc 



a + or f g J" ) ' also S P = V a + 2 6 x + 6 2 , and 



a \ 2 a J 



* Horologium Oscillatorium, ed. 1673, p. 159, App. 



