8 
9. To investigate the equation to a plane under the 
form cc cos a + y cos /3 + z cos y = p. 
Let OQ = p (fig. 4) be the perpendicular from the origin 
upon the plane ABC, making angles a, /3, y with the axes of 
iv, y, 5?, respectively ; join AQ, then, OQA being a right angle, 
OA = OQ sec AOQ, or a = p sec a ; 
similarly, b = p sec /3, c = p sec 'y ; consequently, by substi 
tuting in the equation 
cV y Z 
—t- V + - = L 
a b c 
we get 
w cos a + y cos /3 + % cos y = p, 
the equation to a plane in terms of the perpendicular let fall 
upon it from the origin, and the angles which that perpen 
dicular makes with the co-ordinate axes produced in the 
positive directions. 
It will be observed that neither in this article, nor the 
preceding, are the co-ordinates required to be rectangular ; 
only, in the case of rectangular axes we must have (Art. 5) 
cos 2 a + cos 2 /3 + cos 2 y = 1 ; 
but in the case of oblique axes the cosines, which fix the 
position of the normal to the plane, will be subject to a 
different condition. 
Cor. If with the above equation we compare the general 
form of the equation to a plane 
A x + By + Cz = D, 
D A B C 
we get — = = = ; 
p cos a cos p cos y 
which shew that the constant term bears the same ratio to the 
perpendicular on the plane, that the coefficient of each variable