:}.\l ANAL ) IK <.I:MI:/ i;y [CH. in. 



which is a true equation, since each parenthesis \anishcs 

 >.|. uately ly equations (1) and (2). Hence every point of 

 the line /',/'., is on the locus of equation [!'']. and that 

 locus is therefore a plane. Every algebraic equation of /// 

 first degree in three variables represents a plane. 



215. Equation of a plane through three given points. The 



general equation of the first dearer. 



Ax + By + Cz + D = 0, . . . ( 1 



has only three arbitrary constants, viz. the ratios of tin 

 coefficients. If three given points in the plane are 



then these ratios may be found from the three equation^ 

 Ax l + By l + Cz 1 + D = 0, 

 . I /- 2 + y, + Cfe a + D = 0, ... (2) 



considered as simultaneous. 



In solving equations (2) for the required ratios, two special 

 cases may occur : (a) The value of one of the coefficients 

 may be zero, then the ratios determined must not have that 



coefficient in the denominator. E.g., if D = 0, solution 



A ~K (^ A K 



should not be made for , , , but for , (say). 



/ ^ * ^ M* C/ O 



(b) The equations may differ only by constant factors, then 

 the three equations have an infinite number of solutions. 

 This is explained by the fact that the points are on a straight 

 line, and any plane through the line will pass also through 

 the points. 



216. The intercept equation of a plane. A plane will in 

 general cut each coordinate axis at som. definite distance 



