I v . 1 L YTIC OEOMETR Y [On. Ill 



proj. OMMT - proj. OP, 

 hence proj. OM + proj. ^Of + proj. M*P = proj. OP, 



that is, XCOHa -f I/C080 + 2C08f=l>- . . . [IS] 



'I'll is is called the normal equation of the plane. 



Tin- re are two special cases to be -considered : 



(1) If the plane is perpendicular to a coordinate plane, 



e.g.* to the zy-plane (cf. Art. 210), then 7 = 90, cos 7 = 0, 



and equation [18] reduces to 



zcosa + y cos/3=p. . . . [!'.] 



(-) If the given plane is parallel to one of the coordinate 

 planes, e.g., to the ary-plane (cf. Art. 209); then a= = i>0, 

 7 = 0, and eq. [17] reduces to 



= P . . . . 



218. Reduction of the general equation of first degree to a 

 standard form.* Determination of the constants a, 6, c, p 9 

 a, p, -y. I. Intercept form. In Art. 216 a method has 1 -n 

 indicated for reducing the general equation 



Ax + By+Cz + D = Q . . . (1) 



to the intercept form. Since the points (a, 0, 0), (0, b, 0), 

 and (0, 0, c) are on the plane (1), it follows that the inter- 



cepts are 



D , I> D 



"T "5' ~C' ' ' ' 



II. Normal form. If equation (1) and the equation 

 x cos a -f- y cos ft 4- 2 cos 7 p = . . . 



represent the same plane, then their first members can diffc 



- 



The reduction of this article gives a second proof that the general alge- 

 braic equation of first degree always has for its locus a plane. 



