868 1 VIM'//' ',/ <>Mi:H;V IV. 



Moreover, the trace of the surface on any parallel plane, 

 as z = <i, is given by the r(ju;iti<m 



Az* + By* + 2 Hxy + 2 ( + a(?) x + 2 ( Jlf + a.F ) y 



+ (<7a a + 2 tfa + Jf)=0. . . . (2) 



Now, by Arts. 177, 181, the loci of equations (1) ami 

 (2) are conies of the same species, and with semi-ax s jr,- 

 jM.rtional; therefore their eccentricities are equal, and t In- 

 curves are similar. Hence, all parallel plane sections of a 

 quadric are similar conies. 



226. Species of quadrics. Simplified equation of second 

 degree. As will be seen in the following sections, quad r it- 

 si! r faces may be conveniently classed under four species. 

 For, although different plane sections of any surface will in 

 general be conies of different species, still the general form 

 of the surface may be characterized most strikingly by those 

 plane sections which are ellipses, hyperbolas, parabolas, or 

 M might lines. These species are called, respectively, ellip- 

 soids, hyperboloids, paraboloids, and cones; and each species 

 has special varieties, depending upon the nature of a second 

 system of plane sections. To study these species it will be 

 well to simplify the general equation of second degree as 

 much as possible by a suitable transformation of coordinates.* 



A transformation of coordinates changing to a new 

 rectangular system having the same origin as the old. 1>\ 

 equations [14], will transform the given equation of second 

 degree to 



A'i* + B'y 2 + C'z* -f 2 F'yz + 2 G'xz + 2 ff'xy + 2 L'x 



where A', E\ N f are functions of the nine direction angles 

 Compare with Art 176. 



