KEPLER'S PROBLEM. 27 



Cardan's rule. In this case, therefore, the problem of the 

 comet is reduced to infinite series, or to the arithmetic of 

 sines. If the given point is in the vertex of the curve, that 

 is, in the perihelion, the problem is always resolvable, being 

 reduced to the simple extraction of a cube root ; and this is 

 the case of comets which fall into the sun. 



The resolvable case of the lemniscata is in the same circum- 

 stances, as may easily be seen by inspecting its equation. 



In substituting for J* y d x, its value in our general equation, 

 we may either give it in terms of x, that is, of the abscissa ; 

 or in terms of x y, that is, of the circumscribing rectangle ; 

 and neglect any further substitution. Thence arises a dif- 

 ferent and more elegant solution of the problem, by the inter- 

 section of curve lines ; for we obtain an equation to a new 

 curve, which cuts the former in the point required. Thus, 

 by such a process in the case of the comet, we obtain the 



6mD 2 



equation y = - - ^ -- - to a conic hyperbola. For 

 (m + n) (x + 3 c) 



brevity's sake, put - = d> 2 , the equation becomes y = 

 m -f- n 



3 <i> 2 

 - : Therefore, taking a point on the axis at the distance 



of 3 c beyond the given vertex (or perihelion), erect a per- 

 pendicular, and between the two lines, as asymptotes, 

 describe the hyperbola y x = 3 $ 2 , it will cut the given tra- 

 jectory in the point required: If the given point is in the 

 perihelion, then the perpendicular must be raised at the 

 vertex of the parabola. 



The solution here given by a locus, is evidently general, 

 and has no impossible case. But there are some instances 

 in which such solutions, although perhaps the only practi- 

 cable ones, are nevertheless attended with an impossible case. 

 Let us take that of the lemniscata. Instead of the irresoluble 

 equation of the sixth order, we obtain, by the last-mentioned 



(3 <f - 2 a 2 ) x 



method, a cubic equation ot this form, y = - - _ / - ; 



3 c x x* 2 a? 



