682] 
287 
682. 
FOBMULiE BELATING TO THE BIGHT LINE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 169—171.] 
1. Let X, p, v be the direction-angles of a line; a, /3, 7 the coordinates of a 
point on the line; and write 
a — cos X, f = /3 cos v — 7 cos p, 
b = cos p, g— 7 cos X — a cos v , 
c = cos v, h = a. cos p — /3cos X, 
whence 
a 2 + b 2 + c 2 = 1, 
af+ bg + ch = 0, 
or the six quantities (a, b, c, f g, h), termed the coordinates of the line, depend upon 
four arbitrary parameters. 
2. It is at once shown that the condition for the intersection of any two lines 
(a, b, c, f, g, h), (a, b', g', h'), is a/' + bg' + ch' + af+b'g + c'h = 0. 
3. Given two lines (a, b, c, f g, h), (a\ b', c', /', g', h'), it is required to find their 
shortest distance, and the coordinates of their line of shortest distance. 
Let 
Ax + By + Gz 4- D =0, 
Ax + By + Cz + D' = 0, 
be parallel planes containing the two lines respectively; then the first plane contains 
the point a A r cos X, /3 -t- r cos p, 7 A r cos v, and the second contains the point 
a' + r cos X', /3' + r' cos p, <y' 4- r' cos v ; that is, we have 
Aa + B/3 A Gy A D = 0, 
A a' + B& + G/ + B = 0, 
A cos X + B cos p + G cos v = 0, 
A cos X' A B cos p A G cos v = 0,