346 ANALYTIC GEOMETRY [Cull 



of the third variable^ and having its form and pori- 

 determined by the plane curve represented by the same 



As a direct consequence, it is clear that if a cylinder lias 

 its axis parallel to a coordinate axis, a section made by a 

 plane, perpendicular to that axis, is a curve parallel to and 

 < Mjual to the directing curve on the coordinate plane, and is 

 represented in the cutting plane by the same equation. 

 Thus, the section of tbe elliptical cylinder whose equation is 

 3a^ -f y 2 = 5, cut by the piano z = 7, is an ellipse equal and 

 parallel to the ellipse whose equation is 3s 2 + y* = ;">. 



211. Equations in three variables. Surfaces. A solid 

 figure has the distinctive property that it can be cut by a 

 straight line in an infinite number of points, while a sur- 

 face or line can, in general, be cut in only a finite number. 

 A line has the distinctive property that it can be, in gen- 

 eral, cut by a plane in only one point, while a surface may 

 be cut in a curve. To show that the locus of an algebraic 

 equation in three variables is, in general, a surface, it is suf- 

 ficient to show that, in general, a plane will cut it in a curve, 

 while a straight line will cut it in a finite number of points. 



Let the given equation be 



/Gr,y, 2)=0, (1) 



and let z = <?... (2) 



be a plane parallel to the 2#-plane. The points of inter- 

 section of these two loci will be on the locus of the equation 



and, by Art. 210, they lie, therefore, upon a plane curve, cut 

 from the cylinder whose equation is (3), by the plane whose 

 equation is ('2). Hence the locus of equation (1) is not a line. 



