52 NEWTON S PRINCIPIA. 



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 pro 

 portion is for the circle. Let S be the centre of forces 

 and K of the circle, P T a tangent, S Y a perpendicu 

 lar to it, KM and M P co-ordinates, S K = &, KO = a, 

 P M = ?/, and M K = x. Then, by similar triangles, T K P 



S T x KP 

 and TSY, we have SY= r ,. , or (because the 



sub-tanent M T = , and 2 = ^ 2 +y 2 ) a O r 





( -- : ^a~~J ; a * so ^ P = v /fl 2 + %b x + b* 9 and because 



by the property of the circle O S x S B or (a + b) 

 (a-l&amp;gt;} = a 2 -b 2 = PSxS V; therefore 



SV=- &quot; 2 ~* 2 andPV = 



^ a? + 2bx + b* 



Now by the formula already stated as Bernouilli s, 

 but really Sir Isaac Newton s, the centripetal force in 



SP 



P is as o-^rr - ^ } R being the radius of curvature, 



O JL X JAi 



and in the circle that is constant being = , the semi- 



diameter ; therefore the force is as a * + 2 



8xa*V a * + 2bx+b* . . B O 2 x S P 

 or as j- g , , 3 , that is 2 



B Q 2 x S P 3 



or as /.T. o . ^ 7 \u ^ 02 



BO 



SP 3 



J -J- 9 /&amp;gt; r 



= P V. Therefore the central 



