i UFACK8 W7 



Again, let y &, s c 



be the equations of a straight line (Art. 209), parallel t< 

 avaxifl. The points of intersection of loous (1) and the line 

 vill be also on the loous of the equation 



/(r,*,,)-" 



which, since the equation is in one varial>l< , <>f finite degree, 

 will represent u finite number of planes parallel to the y- 

 plano, aii'l therefore having a finite number of points of 



A nli tin- line ( 1). Hence the locus of equu 

 not a s. 



Therefore, the locut of any algebraic equation in three van- 

 Met it a turf ace. 



212. Curves. Traces of surfaces. Two surfaces intersect 

 in .i . ui\r in space, and since every algebraic equation in 

 solid analytic geometry represents a surface, a curve may be 

 fsented analytically l>\ the t\\. equations, regarded as 

 .simultaneous, of surfaces which pass through it. Thus it 

 has been seen that the equations // '.? = <? separately rep- 

 resent planes, but considered as simultaneous represent the 

 line \vliirh is tlir intersection <>f those planes. Hut 

 he reasoning of An. 11. the given equations of a curve 

 may be replaced by simpler ones \\hich represent other sur- 

 faces passing through the same cu: In dealing with 

 < in ves it is often useful to obtain, from the equations i: 

 equations of cylinders through the same c> t is 

 generally useful t.. represent a curve by two equations each 

 in t\\ \ari.il.les only. 



EXAMPLE : The curve of intersection of the two tarfsoas, 



-25 = mud ('.) x 4 y* - 16 = 0. 



