CHAPTEB II 



THE LOCUS OF AN EQUATION. SURFACES 



208. Attention has been called to the close anal 

 between the corresponding analytical results for the geom- 

 etry of the plane and of space. It is c\i<l. m that in 

 geometry of one dimension, restricted to a line, the point is 

 the elementary conception. Position is given by one vari- 

 able, referring to a fixed point in that line ; and any alge- 

 braic equation in that variable represents one or more points. 

 In geometry of two dimensions, however, it has been sh<>\vn 

 that the line may be taken as the fundamental element. 

 Position is given by two variables, referring to two lix< -<1 

 lines * in the plane ; and any algebraic equation in the two 

 variables represents a curve, i.e., a line whose generating 

 point moves so as to satisfy some condition or law. Conv- 

 spondingly, in geometry of three dimensions the surface is the 

 elementary conception. Position is given by three variables, 

 referring to three fixed surfaces, since any point is the inter- 

 section of three surfaces ; f and it can be shown that any 

 algebraic equation in three variables represents some suit 



* With polar coordinates, these lines are a circle about the pole with 

 radius = p, and a straight line through the pole making the angle with the 

 initial line (Art. 23). 



t With polar coordinates, these surfaces are a sphere, about the origin as 

 center, determined by the radius vector p, a right cone about the z-axis, with 

 vertex at the origin, determined by the angle $>, and a plane through tin 

 *-axis determined by the angle (Art 201). 



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