48 NEWTON S PRINCIPIA. 



Bernoulli!, two years afterwards, wrote his letter to Herr- 

 man (Mem. Acad. des Sciences, 1710), he gives it as his 

 own discovery, and as such it has generally been treated, 

 with what reason we have just seen. He is at much 

 pains to state, p. 529., that he had sent it in a letter to 

 Demoivre in February, 1706 ; but the Principia had been 

 published nineteen years before. Herrman, in his Phoro- 

 nomia, erroneously considers the expression as discovered 

 by Demoivre, Grandi, and Bernouilli. (Lib. I. Prop. 

 XXII.) 



In all these cases p is to be found first, and the expres 

 sion for it (because, pp. 42, 43., TP:PM::TS:SY 



and T S= } an(i PT =. Vdy^ + dx*} is p 



= SY -^y dx - xd &amp;gt;y_^ vy 



~~~ m Also r = 



= N/# 2 -fy 2 . Then the radius of curvature K = 



(dx* + dy 2 ) , . dy . 



j - 3v~ (X being in terms of x, and having no 

 d# 2 xrfX v & d x 



differential in it when the substitution for dy is made). 

 Therefore, the expression for the centripetal force becomes 



***dX . . , 



5 in which, when y and d y are put 



in terms of x, as both numerator and denominator will be 

 multiplied by d # 3 , there will be no differential, and the 

 force may be found in terms of the radical that is, of 

 r, though often complicated with x also. It is generally 

 advisable, having the equation of the curve, to find /?, r, 

 and R, first by some of the above formulas, and then sub 

 stitute those values, or d p and d r, in either of the 



expressions forF, - or 



