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LXVI. On Asymptotic Straight Lities, Plmiesy Cones and Q/- 

 linders to Algebraical Surfaces. By Thomas Weddle*. 



IN the Cambridge Mathematical Journal, first series, vol. iv, 

 pp. 42-4' 7, the late D. F. Gregory gave a very excellent 

 method of determining the asymptotes to algebraical curves. I 

 here purpose considering the corresponding subject relative 

 to algebraical surfaces ; and as this seems to have as yet en- 

 gaged but little attention (if any), I trust the discussion will 

 not be unacceptable to the mathematical readers of this 

 Journal. 



Definitions. 



1. A straight line which passes through a point at a finite 

 distance and touches a surface at an infinite distance, is called 

 an asymptotic straight line, or simply an asymptote to the 

 surface. 



2. 1( eve7-y straight line drawn in a plane be an asymptote 

 to a surface, the plane is styled a conical asymptotic plane to 

 the surface. 



3. If all straight lines drawn in a plane parallel to a straight 

 line in that plane be asymptotes to a surface, the plane is de- 

 nominated a CYLINDRICAL asymptotic plane to the surface. 



4. An asymptotic cone or cylinder to a surface is a cone or 

 cylinder having its generators asymptotes to the surface. 



If <Pq{xyz) denote a homogeneous function of a;,y, z of the g'th 

 degree, it is plain that a surface of the p\h degree may be 

 denoted thus: 



Let 



x—u V—B z—y , , ^ 



I m n ^ ' 



be the equations of an asymptote to (1.) passing through the 

 point (a/3y) : hence 



x—lr-\-a, j/=wr + /3, and ;^ = 7M- + 'y ; 



substitute these values oi x,y and z in (1.) and develope each 

 term, the result is, 



* Communicated by the Author. 



t The axes may be either rectangular or oblique ; only in the former 

 case we shall have 



but in tlie latter, 



/,g, h denoting the cosines of the angles which the axes make with each 

 other. 



