KEPLER'S PROBLEM. 25 



given ratio of m to n. Let x and y, as usual, be the abscissa 

 and ordinate, and c the eccentricity of the given point, through 

 which the radius vector is to be drawn, if the equation is taken 

 from the centre ; or, if it is taken from the vertex, let c be 

 the distance of the given point from that vertex, as the focal 

 distance in the case of the planets or comets (supposing the 

 comets to revolve round the sun in parabolas or eccentric 

 ellipses, having the sun in the focus), then, it may easily be 

 found, that the following differential equation 2 J* y d x + y 



(c x) = , if resolved for the case of any given curve, 



m + n 



gives a solution of the problem for that curve. Instead of 

 fydx, there must be substituted the general expression for 

 the area found by integration ; and y must then be expressed 

 through the whole equation in terms of x, or x in terms of y : 

 There will result an equation to x, or to y, which, when re- 

 solved, gives a solution of the problem. 



Now, it is manifest, that one or both of two difficulties or 

 impossibilities may occur in this investigation of the value 

 of x. It may be impossible to exhibit f y d x in finite terms ; 

 and it may be impossible, even after finding "T y d x, to resolve 

 the equation that results from substituting the value of f y d x 

 in the general equation above given. Thus, if the given 

 curve is not quadrable, the equation can never be resolved ; 

 but, although the curve is quadrable, it does not follow that 

 the equatit .1 can be r~ solved. 



In the case of the circle and ellipse, both these difficulties 

 must of course occur The value of f y dx in the circle being 



nd x 



J dx V a x x 3 , and in the ellipse - /J ax a? (where 



a and b are the transverse and conjugate), neither of which 

 differentials can be integrated in finite terms, the general 

 equations become indefinite or unintegrable. 



The lemniscata (a curve of the fourth order) is quadrable in 

 algebraic terms : but the resolution of our general equation 

 cannot, in this case, be performed in finite terms ; it leads to 



