By Henry Parker Manning

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We may call it a surface of double revolution. The parallel circles about one axis are meridian curves for the other axis. If a solid body, or, we may say, all space, move along a straight line without rotating about it, it will rotate about the conjugate line as an axis without sliding along it. A motion along a straight line combined with a rotation about it is called a screw motion. A screw motion may then be described as a rotation about each of two conjugate lines or as a sliding along each of two conjugate lines.

I, I, 5), Now we can let O move off on C M , the construction remaining the same. That is, we let the lines C A and C B rotate about C without changing their lengths, in such a manner that the three perpendiculars D O, E O, and F O shall always pass through O. As O moves off indefinitely, the angles a b at C approach Π and Π as limits, and the three perpendiculars 2 2 approach positions of parallelism with C M and with each other. But the triangle A B C approaches as a limit a triangle which is equal to ABC, having two sides and the included angle equal, respectively, to the corresponding parts of the latter.

The distance, δ, between two points: cos iδ = cos iρ cos iρ + sin iρ sin iρ cos(θ − θ). δ and one of the points being fixed, this may be regarded as the polar equation of a circle. CHAPTER 5. ANALYTIC NON-EUCLIDEAN GEOMETRY 58 3. The equation of a line: Let p be the length of the perpendicular from the origin upon the line, and α the angle which the perpendicular makes with the axis of x. From the right triangle formed with this perpendicular and ρ we have tan iρ cos(θ − α) = tan ip. This is the polar equation of the line.