### Quadratic systems of circles in non-euclidean geometry

D. M. Y. Sommerville
1919 Bulletin of the American Mathematical Society
1919.] QUADRATIC SYSTEMS OF CIRCLES. 161 For an even X, this becomes a -OL<L + #4 -• • • = 1. For any X, there is a\ -a 3 + OJ 6 -• • • = 0. When X is odd, then -a + OL% -a\ + • • • = 1. When n is odd, say n = 2X + 1, then a -«2 + «4 ~ • • • = 0, and a% -az + «5 -• • • = 1, when X is odd; -ai + a 3 -as + • • • = 1, when X is even. We shall, in particular, consider the case where (15) has the form (16) in which n and A must both be either even or odd in order that (16) may reduce to (14) for u =
more » ... -1, v = -*i. After a rather complicated process of elimination the cartesian equation of this special class of curves with the nth. roots of unity as foci becomes which is an n-ic. For n = 3, k = 1, we get the cubic hyperbola xy 2 = -4/27. The condition for a proper n-ic in (17) is evidently n -2k ^ 1, n ^ 3. § 1. The equation of any circle can be written (1) kS-a 2 = 0, where S = 0 is the equation of the absolute, and a = lx + my + nz = 0 is the equation of the axis, the center being the absolute polar of this line.