EQUATION OF THE SKCOND DEGREE. 197 



there can be no intersection of compound ordinates, and 

 these can have no envelope. 



The three equations (a), (/3), (7) possess this remarkable 

 reciprocal property. I£ any one of them be taken as the 

 equation to be represented_, it is itself the Cartesian equa- 

 tion of the corresponding curve when x^ and y* are both 

 positive; and the two others are Cartesian equations of 

 the respective envelopes when either x^ or \f is alone 

 negative ; and thus the ellipse (a) and the hyperbola (/8) 

 are the representative curves of all the three equations. 



The three similar equations {ci), {^'), (7'), in which the 

 only diflference from the former three is in the reversed 

 sign of unity on the right-hand sides^ possess, of course, 

 the same reciprocal property. But here the impossible 

 curve is on the xy plane ; that on the xz plane is the con- 

 jugate hyperbola to the former one, and is on the same 

 plane with it, and has therefore the same asymptotes, 

 whilst the curve on the yz plane is an ellipse whose semi- 

 axes are a and {ai^ — b^) ^. 



When a = b — that is, {a) is a circle — the three other real 

 curves coincide with the z axis. 



To find the Cartesian equation to the envelope of the 



circular ordinates in the case No. 2, in which y'' alone is 



negative, we have, for the positive square of the type radius 



b^ 

 of the circles, — (.27* — a*), and, for the equation to a type 



circle, 



(f-^)^+r=^.K-«*), (I) 



where x is the variable parameter ; hence, taking the first 

 derivative, 



-{^-^) = ~.^, (2) 



