240 CENTBAL FORCES; 



values ^ in terms of x and y, we obtain a mean of estimating 

 the force as proportioned to r, which is V x 1 4. y*. 



5. The last article affords, perhaps, the most obvious me- 

 thods of arriving at central forces, both directly and inversely. 

 Although the quantities become involved and ernbarassing in 

 the above general expressions for all curves, yet in any given 

 curve the substitutions can more easily be made. A chief 

 recommendation of these expressions 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- 



d s 8 



times erroneously given for the centripetal)* is -, s being 



2 ri 



the length of the curve and E the radius of curvature. This 

 gives the ready means of working if that radius is known. 

 But its general expression involves second differentials, the 



ds 3 

 usual formula for it being /d y\; consequently we 



Ct Ou s\ d I p I 



\dxj 



dy 



must first find = X (a function of x}. and then there are 

 d x 



only first differentials. 



rfs 2 

 Another for this radius of curvature is 



r d T 



and this is used by Laplace ; and another is , which, with 



dp 



other valuable formulas, is to be obtained from Maclaurin's 

 Fluxions. But the formula generally ascribed to John Ber- 



* This error appears to Lave 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. 



