44 NEWTON S PRINCIPIA. 



parallel to the co-ordinates S M, M P. Then P being 

 the middle of the arc, twice the triangle S P Q is propor 

 tional to the time in which C Q is described. Therefore 

 QP x SYorQL xPSis proportional to the 



C Q 



time ; and Q R is the versed sine of &amp;lt;^, therefore 



Q O T? 



F as r -T 2 becomes F as T~9 an( ^ if S M = a:, 



T 



M P = y } and because the similar triangles Q K o and 



S M P give Q R - ^f^&amp;gt; and because A M being 



O -A-L 



the first differential of S M, o Q is its second differential 



(negatively), therefore Q R = - x &quot;*&quot;^ (taken 



x 



with reference to d t constant), and F is as 

 ~f^,y*,,- But LQ 2 = QP 2 - LP 2 and 



L P is the differential of S P or V x * + f. Therefore 



2 



4 ( d-\ 

 (x d y v d x} 2 &quot; \ y) 



L Q 2 = v ^ y } = 2 ^2~, and F is as 



x 2 + 2 x 2 + 2 



y* 



But as the differential of the time (L Q x P S) may 



be made constant, Q R will represent the centripetal 

 force ; and that force itself will therefore be as 



*L. ) * taken with reference to d t constant. 



* 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 ____Jtll i s without integration an useful one. 



* y 



