286 
MR. R. S, HEATH OH THE DYNAMICS OF 
8. The angle between two lines is equal to the distance between their poles ; i.e., if 
6 be the angle between two lines whose poles are (/, m, n), (l', m, ri), 
cos 6=11' mm' nn . 
To draw a perpendicular from a point to a line, we have only to join the point to 
the pole of the line. In general we can draw only one perpendicular from the point 
to the line, but if the point be the pole of the line, every line through it is a 
perpendicular to the given line. Let or be the sine of the perpendicular from a point 
( x, y, z) to a line 
lx J r my J rnz=0. 
Then 07 denotes the cosine of the distance of the point from the pole of the line, 
therefore 
or = lx-\-my-\-nz. 
The equation to the line joining two points (x, y', z), ( x", y ", z") is 
x y z 
x y z 
ft // A 
x y z 
= 0 
The equation to a line drawn from a point (x', y', z') perpendicular to a line 
lx J r viy-\-nz = Q is 
y z =0 
x 
/ / / 
x y z 
l m n 
9. We now proceed to establish the Trigonometry of any plane. Let A B C be a 
triangle, and, for simplicity, let C be an angle of the triangle of reference, and let A, B 
be the points 1, 2. Denoting the sides of this triangle by a, b, c, we get 
cos c=x l x i ,-\-y 1 y. 2 -\-z l z. 2 
cos a=z 2 
cos b=z 1 
Again, the equation to the absolute pair of lines through C is 
x 2 ~hy 2 =0. 
It is easy to see that the formula 
cos 
TT 2 
2 g— U 12- 
U n U,., 
