45 

 3. The rectangle S Y x Q P being equal to Q L x S P 



and s Y = yd^^Ly we have F as 



Vdx 2 + dy* 

 QR QR QR 



OR OP 2 



4. Because F = g ^ 2 ^ Q p 2 and -- is equal to the 



chord P V of the circle, which has the same curvature 

 with Q P O in P, and whose centre is K (and because 

 Q P 2 = Q R x P V by the nature of the circle and 

 the equality of the evanescent arc Q P with its sine, and 



thus P V = ^J 2 , - therefore ^ = ^ ), Fisas 



Q y 2 - p~v* I n like manner if the velocity, which 



v 2 

 is inversely as S Y, be called v, F is as py. Now the 



chord of the osculating circle is to twice the perpendicular 

 S Y as the differential of S P to the differential of the 

 perpendicular; and calling S P the radius vector r, and 



S Y=, we have PV = ~ -, and F is 



dp 



Q 7 



also F is as . In these formulas, substituting for p 



and r their values in terms of x and y, we obtain a mean 

 of estimating the force as proportioned to r, which is 

 V x*+y 2 . 



5. The last article affords, perhaps, the most obvious 

 methods of arriving at central forces, both directly and 

 inversely. Although the quantities become involved and 

 embarrassing in the above general expressions for all 

 curves, yet in any given curve the substitutions can more 

 easily be made. A chief recommendation of these expres- 



