57 



Some Skew Surfaces of the od and 4th Degree. C. A. Waldo. 



LAbstnict.J 



The theory of skew ruled surfaces has been specially studied by Cremona, 

 Cayley, Eohu and others. Rohn of Dresden has contributed several imjiortant 

 series of models of general and fundamental character. 



The object of this paper is to discuss somewhat in detail by Cartesian 

 coordinates a family of surfaces formed by a straight line generatrix moving 

 along two non-intersecting straight lines and a plane curve whose plane is parallel 

 to both right lines. 



Let the plane curve be f ( m, n ) = Aui^ + Bm^ i -f Cm"^"' -\- .... L = 0. 

 Let the orthogonal projections of the straight lines on the plane of this curve be 

 axes of X and Y, and their common perpendicular the axis of Z. Let the 

 distance from the plane of the curve to one right line directrix be pb, to 

 the other qb, the directrix parallel to the Y axis being the more remote. In this 

 position, by similar triangles, it is easily shown that m : x :: pb : pb— ^z, and 

 n : y : :qb: z— qb. Substituting these values in f(m, u)^0 we have at once a 

 general expression for the Cartesian equation of an unlimited number of skew 

 surfaces of this description, viz. : 



^ ji^}\ b\j^Y: 1^1+ .... L=o 



(pb — zj (pb — zj (z — qbj 



As shown by Salmon in another way the degree of this surface is at once seen 

 to be twice that of the directing curve or twice the product of the degrees of the 

 directing lines of the surface. 



Plane sections of this surface are in general of the 2Kth degree, but when 

 made by the plane Z=constant, they degenerate to the Kth degree. 



If the directing curve be of the 2d degree the resulting surface will be of 

 the 4th degree unless degraded by some special position. If we take the circle as 

 our curvilinear directrix and place it half way between the two rectilinear 

 directrices the resulting equation will be of the form 

 b'' x^ b^ y^ _ , 



(b — z)2 + (b+z)2 ~ ''" ^ '' 

 If the circle be replaced by the equilateral hyperbola we have 



(b-z)2 (b-fz)2"~^ ^^^• 

 If the directing curve be the parabola, x n2=4pm, the surface is 

 b y2 p X 



(b + z) 

 a surface of tiie 3d degree. 



(3), 



