62 
Of equation (a) we have the following cases, a? and b 2 being 
always positive numbers : — 
Case 1. When x 2 and y 2 are both positive numbers the 
ordinates are both tubular, and the equation is represented, 
as in the Cartesian system, by the ellipse (a) whose semi» 
axes are a and b. 
Case 2. When y 2 alone is negative, that is, when x 2 is 
positive and greater than a 2 : — the x ordinates are tubular 
and the y ordinates circular, and the equation is represented 
by an envelope on the xz plane, the Cartesian equation to 
which we shall presently show to be (/3), viz : — That of the 
hyperbola Q3). This hyperbola has its vertices in the foci 
of the ellipse (a), and its foci therefore at the vertices of the 
ellipse, and its transverse axis is of the same magnitude as 
that of the ellipse. 
Case 3. When x 2 alone is negative, that is, when y 2 is 
positive and greater than b 2 : the y ordinates are tubular 
and the x ordinates circular : the envelope of the latter has 
(y) for its Cartesian equation, and is an impossible curve 
or in other words, there is no intersection of the circular 
ordinates and therefore no envelope. 
