MATHEMATICS: A. EMCH 
223 
A much favored method, especially in descriptive geometry, consists in 
considering ruled surfaces as continuous sets of straight lines, or generatrices, 
which intersect three fixed curves, the directrices, simultaneously. If these 
are algebraic curves of orders /, m, n, with no common points, the ruled 
surface which they determine is, in general, of order 2 I. m. n. 
Frequently, ruled surfaces are also defined as systems of elements, either 
common to two rectilinear congruences, or to three rectilinear complexes. 
Of great importance for the following investigation is the definition of ruled 
surfaces as systems of lines which join corresponding points of an (a, ^) — cor- 
respondence between the points of two algebraic curves and C„ of orders 
m and n. If these curves are plane, and if to a point of Cm correspond a 
points on C„, and to a point of C„ ^ points of Cm, then the order of the surface 
is in general a m -\- ^ n. 
Finally there is the cinematic method in which ruled surfaces are generated 
by the continuous movement of the generatrix according to some definite 
cinematical law. In this connection the description of the hyperboloid of 
revolution of one sheet is well known. 
The literature seems to contain but little about this method of generating 
ruled surfaces. A number of treatises on differential geometry contain chap- 
ters on cinematically generated surfaces. 
The class of surfaces here considered is obtained as follows: Given a directrix 
circle C2 and a directrix line Ci, which passes through the center of C2 at right 
angles to the plane of C2. The generatrix g moves in such a manner that a 
fixed point M oi g moves uniformly along C2, while g in every position passes 
through Ci. The plane e through Ci in which g lies evidently rotates about 
Ci with the same velocity kd as M. In this plane g rotates about M with a 
uniform velocity kiid. When = p/(l '^^ a rational fraction, the surface 
generated is also rational and belongs to the class of ruled surfaces generated 
by means of an (a, /3) correspondence between Ci and Co. 
When Ci coincides with the 2-axis, so that C2 lies in the xy-plane, and we 
denote by p the distance of the projection of a point P on the generatrix g 
from the origin and by d the angle P'OX, the equations of the surface ex- 
pressed by the parameters p and 6 are 
P 
X = p cos 6, y = p sin 0, z = (p — a) cot - 6. 
It is shown that these may be expressed rationally by p and another param- 
eter /. Also the implicite cartesian equation of the surface is obtained, as 
well as are the parametric equations of the double curve of the surface. The 
following theorems are of interest: 
Theorem 1. The surface of the class is rational and of order 2 (p -\- q) or p 
q, according as q is odd or even. 
Theorem 2. When q is odd the generatrices of the surface cut Ci and C2 in two 
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