422 APPENDIX. 



Ay 



for the equation becomes I 2 f 2 dx = 



= _ _. He makes no mention of this 



y 2 (2 B - log y 2 )2 



case, as, like Herman and most others, he seems unwilling 

 to approach it ; undoubtedly, however, such is the applica 

 tion of his formula. 



Keil, in his paper on Central Forces in Phil. Trans. 1708, 



p. 174., gives the case of the force as- and reduces it to 

 finding _ P, the perpendicular to the tan 

 gent. 



By one process grounded on Prop. XLI. Lib. I., this 



result is obtained for the case of -, that is - 





2 dxd? x + 2dy d* y _r (x d x + y d 



y 



or 



//- 



, ^ 



/d x 2 + d ?/ 

 j-p 2 log (&amp;gt; 2 +y 2 ) -c = 0; and d t* being 



_ (y d x x d y) 2 s* d x* + d ?/ 2 



72 &amp;gt; tne equation becomes / =-4-_ 



J (ydxxdyf 



The process grounded on the formula / = h 



2 p .xv 



is, if possible, more hopeless; for this gives 



h (?/ 2 + (x - c}*f X (d x d 2 y - d y d 2 x) 

 Z(ydx xdy-icd y) 3 



1_ 



, or h (y* + (x c) 2 ) (d x d 2 y 



