292 
MR. R, S. HEATH OX THE DYNAMICS OF 
r 
The quantities a, b, c,f g, h, are thus reduced to four independent variables and are 
called the six coordinates of the line. 
16. If we interchange the letters ( x, y, z, u ) with the letters (l, m, n,p), we get the 
polar line of the first. Hence the coordinates of the polar line are 
f 9, h, a, b, c. 
The co-ordinates of the line joining two points ( x, y, z, u), (x\ y, z', u') distant an 
angle 6 from each other, are 
sin 9 
&c. 
Similarly for the line of intersection of tw r o given planes. 
It has been incidentally proved that the conditions that a line a, b, c, f g, h should 
lie in a plane l, m, n, p, are 
— cm-\-bn — fp=0 j 
cl . —an — gp=0 I 
— bl-^-am . — hp =0 | 
fl -\-gm-\-hn . = fij 
which are equivalent to two independent relations. These are also the conditions that 
the polar line f g, h, a, b, c should pass through the point (/, m, n, p). 
The coordinates of a line through a point {x, y , z, u) perpendicular to a plane 
(l, m, n, p) are 
a — 
yn—zm 
sin 9 
&c. 
where 9 is the angle between (x, y, z, v) and the point (/, m, n, p), which is the pole of 
the plane. 
17. We shall now find the length of the perpendicular from any point (x, y, z, it) to 
a line a, b, c,f g, h. 
Let (l, m, n, p), (l\ m, n', p’) be two perpendicular planes passing through the line. 
Let Tiq, ct, be the sines of the perpendiculars from (x, y, z, u) on these planes, and 
let trr be the sine of the perpendicular on the given line. Then by Spherical 
Trigonometry 
o_ o . 2 
7TT- 777 | ' 777 i' • 
7 7T i = lx -f my -f nz -\-pu 1 
zs 2 z=l'x-\-m'y-\-nz-\-p'u I 
Now 
