198 MR. CHARLES CHAMBERS ON THE 



and eliminating x by means of (i) and (2), we obtain 

 — ^ =1, or — — r-— ^=1, 



which is the equation (/3) written above. Similarly, by 

 dealing with the impossible ordinates to the y abscissae, we 

 obtain the equation (7) . 



The same three equations (a), (/3), (7) have the reciprocal 

 property that has been described when, instead of the axes 

 being rectangular, the primitive equation is referred to a 

 pair of conjugate diameters as axes, provided, however, 

 that the tubular ordinates be o£ elliptical transverse section 

 such that their section by the plane of the two other coor- 

 dinates is circular. 



The extension of the method so as to represent an 

 equation between three variables, referred to rectangular 

 axes, presents no difficulties. 



It will be observed that the envelope of the circular or- 

 dinates is the same central conic that we should obtain by 

 taking, in the proper plane, ordinates affected with the 

 sign \/— I as normals instead of ordinates, and measured 

 from the extremities of the respective abscissae which yield 

 them. 



One interesting consequence of taking account of both 

 the representative curves of an equation is that it affords 

 a geometrical illustration of the four-point intersection of 

 pairs of conies in cases which are unintelligible without it ; 

 in the case of similar and similarly situated concentric 

 ellipses, for example, the absence of any points of inter- 

 section on the ellipses is compensated for by the touching 

 of the pair of supplementary hyperbolas on the coincident 

 asymptotes ^. 



* The writer is under the impression that he will find that, in the general 

 case of elimination between the equations of a pair of conies, if one or two 



