46 NEWTON S PEINCIPIA. 



sions is, that they involve no second differentials, nor any 

 but the first powers of any differentials. But it may be 

 proper to add other formulas which have been given, and 

 one of which, at least, is more convenient than any of the 

 rest. 



One expression for the centrifugal force (and one some- 



7 o 



times erroneously given for the centripetal) * is ^p, s be- 



-t _Lv 



ing the length of the curve and R the radius of curvature. 

 This gives the ready means of working if that radius is 

 known. But its general expression involves second diffe- 



ds* 



rentials, the usual formula for it being ~ fdv 



2 x d -=^ 



consequently we must first find - = X (a function of 



CL X 



x), and then there are only first differentials. 

 Another for this radius of curvature is 

 ds* 



is Used ^ Laplace ; and ano- 



V fl 7* 



ther is -5 -, which, with other valuable formulas, is 



to be obtained from Maclaurin s Fluxions. But the for 

 mula generally ascribed to John Bernouilli (Mem. Acad. 

 des Sciences, 1710), is, perhaps, the most elegant of 



7* 



any, F = - - 5 - ^; and this results from substituting 



A . p X -tt a 



2 T d T 



2 R for its value i - , in the equation to F, deduced 



above from Newton s formula, namely, F = ^ . 



2p 3 dr 



* This error appears to have arisen from taking the case where the 

 radius of curvature and radius vector coincide, that is, the case of the 

 circle, in which the centrifugal and centripetal forces are the same. See 

 Mrs. Somerville s truly admirable work on the Mec. Cel., where the error 

 manifestly arises from this circumstance. 



