A \. I /.I"/'/'- (.1. [Cii. II. 



is also tlu- iiii.-rsoction of the surfaces 



* + y + s*-25-(z+y-16)=0, i.e., *=3, (a) 



with the surface (2). The curve is therefore composed of two circles of 

 radius 4, parallel to the zy-plane at distances + 8 and - 3 from it. 



Conversely, the curves of intersection of a surface with 

 tin- courdinati' planes may be used to help determine tin- 

 nature of a surface. These curves are called the traces of 

 the surface. 



Thus, the surface z 2 + / + z 2 = 25 has the traces 



on the yz-plane, where x = 0, y 2 -f z 2 = - "> ; 

 on the or-plane, where y = 0, x 1 -f z 2 = 25 ; 

 on the a-^-plane, where z = 0, z 3 -f- y 2 = 25. 



li of these traces is a circle of radius 5, about the 

 origin as center; the surface is a sphere of radius 6 \\ith 

 center at the origin. 



Since three surfaces in general have only one or more 

 separate points in common, the locus of three equations, <-.u- 

 sidered as simultaneous, is one or more distinct points. 



213. Surfaces of revolution. Analogous to the cylinoYrs 

 are the surfaces traced by revolving any plane curve about 

 a straight line in the plane as axis. From the method of 

 formation, it follows that each plane section perpendicular 

 to the axis is a circle, the path traced by a point of the 

 generating curve as it revolves; and the radius of the < ir< lc 

 is the distance of the point from the axis in tin- plane before 

 revolution begins. These facts lead readily to the equation 

 of any surface of revolution, as a few examples will show. 



(a) The cone formed by revolving about the z-azis the line 



2* + 3* = 15. (1 



