4 
but the triangle OAN, right-angled at A, gives 
ON 2 = O A 2 + AN 2 ; 
OM 2 = OA 2 + JA 2 + NM 2 ; 
d = \/x' 2 ■+• y 2 + z' 2 . 
Cor. Let a, /3, be the angles which Oil/ makes with 
the axes of x, y, z ; then, joining AM, since angle OAM is 
a right angle, 
cos a = 
OA 
OM 
x 
d ; 
x' = d cos a ; similarly, y = d cos ¡3, z = d cos y. 
<Xj* ^ 'll ^ -L % ^ 
Also (cos a) 2 + (cos (3) 2 + (cos y) 2 = — = 1. 
ct 
This is the condition which the three angles, made by 
a line through the origin with the co-ordinate axes, must 
satisfy ; they cannot, therefore, be assumed at pleasure, but 
two being given, a, ¡3, for instance, the third is determined 
by the equation 
cos y = ± \Z"I - (cos a)~ - (cos /3) 2 , 
and will therefore have two values, y and tt — y ; the value 
7r — y corresponding to a line 031' in the plane zOM, which 
is inclined at an angle y to Oz, and visibly makes angles 
equal to a, ¡3 with Ox, Oy, the same as OM does. 
6. To find the distance between two points in terms of 
their co-ordinates. 
Let M' (fig. 3) be a point whose co-ordinates are x, y , z, 
and M any other point whose co-ordinates are x, y, z; join 
MM' = d, and upon AIM', as diagonal, describe a rectangular 
parallelepiped having its edges parallel to the co-ordinate axes. 
Then from the triangle M'AIK, right-angled at K, 
MAI' 2 = M'K 2 + KAP = M’K 2 + {z - z'f.