64 
-(Z~ x ) = ~i x (2) 
W 
and eliminating x, by means of (l) and (2), we obtain 
P £ 2 ■, a 2 z 2 
— — — _ l- = 1 or = — = 1 
a 2 - 6 2 6 2 « 2 - 6 2 6 2 
which is the equation (/3) written above. Similarly, by 
dealing with the impossible ordinates to the y abscissse, we 
obtain the equation (y). 
The same three equations (a), (j3), (y), 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, providing, however, that 
the tubular ordinates be of elliptical transverse section such 
that their section by the plane of the two other coordinates 
is circular. 
The extension of the method so as to represent an equa- 
tion between three variables, referred to rectangular axes, 
presents no difficulties. 
It will be observed that the envelope of the circular ordi- 
nates is the same central conic, that we should obtain by 
taking, in the proper plane, ordinates affected with 
the sign V — l 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 conics 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 conics, if 
