EQUATION OF THE SECOND DEGREE. 195 



y= -j-flj. In this way we avoid altogether the introduction 

 of the algebraical symbol V — i as affecting the geometrical 

 representation of the equation. In the case of circiilar 

 ordinates it is obvious that two points in the curve — that 

 is, in the envelope of a series of circles whose centres are 

 on a straight line — have the same ordinate and abscissa, 

 one point, namely, on each side of the straight line. 



The writer is prepared to have what he has advanced so 

 far, regarded as rather curious than instructive, unless it 

 can be shown that the system he describes leads to inter- 

 esting results. In what follows it will be shown that such 

 results are not wanting. 



We now write down a series of equations, and construct 

 a figure to represent them; and we then proceed to comment 

 on the equations and curves, and on their points of interest 

 in connexion with what has gone before. 



Of equation (a) we have the following cases, a* and b^ being 

 always positive numbers : — 



Case I. When x^ and y^ 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* alone is negative — that is, when x^ is 

 positive and greater than a*. The x ordinates are tubular, 

 and the y ordinates circular; and the equation is represented 



o2 



