296 Prof. Sylvester on Periodical Changes of Orbit, 8fb. 

 tesian oval, we have 



-m> 



1 £ v *-9,k*~ 



2 df V ~ M df 



where - is the instantaneous area, and,if the equation to the oval 



be/— kg — m, 



df=kdg; du = {l+k)dg-, dv={l-k)dg; u -( l + ] c jf-'p 



so that 



/1\ 2 _ (u-v)({l + k)%c*-v 2 ) + (1 -k)*(c*-u 2 )) 



\2p) ~ (^^ + ^)((l-^^)(c 2 -^) + (l-^)(c 2 -^)) 2, 



from which F may be calculated and expressed under the form 



P 



t,;^, where P and Q are each rational integral functions of the 



fourth degree in/. 



It does not seem to me worth while to work out the actual 

 values of P, Q for the general form of the oval (in algebra as in 

 common life, there is wisdom in knowing where to stop); but it 

 did appear to me desirable to ascertain the form of the expres- 

 sion for the retaining force, which, it is hardly necessary to add, 

 it would have been quite impossible to do had the ordinary sys- 

 tems of coordinates been employed. The fact of this force being 

 a rational function of the distance is a result not without interest ; 

 and for particular varieties of the curves belonging to the class 

 of Cartesian ovals, it will be easy to obtain its actual value as a 

 function of the distance. 



Postscript, 



On the Curve in Space which is the Analogue to the Cartesian 



Ovals in piano. 

 By a Cartesoid we may understand a surface such that a linear 

 relation exists between the distances of any point in it from three 

 fixed points in a plane, and by a twisted Cartesian the intersection 

 of two Cartesoids whose three fixed points of reference are iden- 

 tical. A twisted Cartesian, then, will be a curve in space whose 

 distances from three fixed points (its foci) are connected by two 

 linear relations : from this it is obvious that it may be conceived 

 also as the intersection of two surfaces of revolution generated 

 by the rotation about their lines of foci of two plane Cartesians 

 having one focus in common, so that it will consist of a system 

 of closed rings. If F, G, H, K be any four points in a plane, 



