858 ANALYTIC GEOMETRY [Cii. MI. 



as the angles between the respective normals from the origin 

 to the planes. If 



Ayi + B#+C l z + D l = 0, . . . (1) 

 and A# + B# + C# + D, - 0, . . . (2) 



be two planes, then the direction cosines of their normals 

 are respectively (eqs. [21]) 



C08fli 



etc. 



and by equation [10], if be the angle between the two planes, 

 and hence between the two normals, 



There are two cases of special interest. 



I. Parallel planes. If the planes (1) and (2) are parallel, 

 their normals from the origin will have the same direction co- 

 sines, and differ only in length ; therefore, by equations [-1 ], 

 the equations of the planes must be such that the coefficients 

 of the variable terms are the same in the two equations, or 

 can be made the same by multiplying one equation by a 

 constant. In other words, if the planes (1) and (2) are 

 parallel, then 



= = '; . . . [23] 



AS B t C 2 



and the plane Ax + By -f Cz + 7T- ... (3) 

 is parallel to the plane 



Ax + By + C* -f D 0, . . . (4) 

 for all values of the parameter K. 





