( II M'll.i; IN- 



EQUATIONS OF THE SECOND DEGREE 

 me 1C SURFACES 



225. The locus of an equation of second degree. The most 

 general algebraic equ.it i*>n of M- -on. I degree in three variable* 



U written 



+ By + CM* + t f V* + 1 Oxn + t HJT* + f Lac + f M y 



which is tin- IINMIM of an equation of 

 degree is called a quadric surface, ami is of particular 

 interest because lone connection with and analogy to 



the conic sections. In i \ plane section of a quadric 



is a conic, as may be easily shown as follows. 



By A rt. 207, any plane may be chosen as a coordinate plane, 

 and the transformation of coordinates to the new axes uill 

 leave the degree of equation [ '.! ) nn. han^- the new 



equation of the locus will still IH> of the form [31], th< 

 with different values for the c* To find the nature 



section, choose the pvcii plane as (say) the 

 plane of reference, and transform to the new axes ; the new 

 equation will le of form (1 . M let i 0. The equa- 

 tion of the section of the quudric is 



AJ* + fly 1 + -J //ry + _ Ls -r '2 My + JT- ; . . (1) 

 and this, by Art. 175, represents a conic. 



