9MOMXTBT [<.. in. 



(Art. -in, and two equations considered as simultaneous 

 ivpivsL-ni the intersections of their two loci (Art. 39). 

 Therefore since two planes intersect in a straight line, the 

 locus of tin- two simultaneous equations of first degree, 



Ap+Btf + Cf + D^Qi Ajc + Btf + Cf + D^Q,. . . (1) 



is a straight line. As suggested in Art. 212, it is generally 

 more simple to represent the straight line by equations in 

 two variables only, standard forms, to which equation (1) 

 can always be reduced. 



222. Standard forms for the equations of a straight line. 

 (a) The straight line through a given point in a given direc' 

 Let P l = (a-p y r Zj) be a given point, and , j3, 7 the direc- 

 tion angles of a straight line through it. Let P = (#, y, z) 

 be any point on the line, at a distance d from P r Then by 

 equation [7], 



= z Zj, d cos/9 = y y^ dcosy = z z r . . . (1) 



hence ^*l = ?L^l = ^l^. . . . [ 26 ] 



COS a COS? COB? 



which are the equations of a straight line in the first standard 

 form, called the symmetrical equations. 



(b) The straight line through two given points. Let P l = 



(*! #1' z i) and A =0*2' Vv z i) be tlie g iven points. Any 

 straight line passing through P l has [26] for its equations. 

 If the line passes also through P y then 



cos cos 7 



and hence from equations [26] and (2), by division 

 eliminate the unknown direction cosines, 



[27] 



