LAW OF THE UNIVERSE. 239 



cl? 3C A/ iJC* "\^ 1J* 



that force itself will therefore be as * taken 



x 



with reference to dt constant. 



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



ydxxdy QR 



and S Y = , , we have F as 



(ydx- xdyj / x 



Cl 



S Y 2 x Q P a 



Q R QP 2 



4. Because F = T =T -- ^ and -77^ is equal to the chord 

 S Y 2 x Q P Q K 



P V of the circle, which has the same curvature with Q P O 

 in P, and whose centre is K (and because QP 2 = QRxPV 

 by the nature of the circle and the equality of the evanescent 



Q P 2 O T? 



arc QP with its sine, and thus P V = , therefore ^r^a 



= ], F is as ^^ ^P^V' ^ n ^ e manner ^ * ne velocity, 



v 2 

 which is inversely as S Y, be called v, F is as ^ . 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 per- 

 pendicular ; and calling S P the radius vector r, and S Y = p, 



2pdr . _. . dp 



we have P V = -~ , and F is as .'' ; and also F is as 

 dp 2p A dr 



e u (I T) 



-. In these formulas, substituting for p and r their 

 2 d r 



* Of these expressions, although I have sometimes found this, which 

 was first given by Herrman, serviceable, I generally prefer the two, 

 which are in truth one, given under the next heads. But the expression 



first given - -- 2" 1S without integration an useful one. 



