288 
FORMULA RELATING TO THE RIGHT LINE. 
[682 
which last equations may be written 
A a + Bb + Cc = 0, 
Aa! + Bb' + Gc' = 0, 
giving 
or, if we write 
and assume, as is convenient, 
then 
A, B, G = 
A : B : G = bc' — b'c : ca' — c'a : ab' — a'b, 
6 = aa' + bb' + cc', 
A 2 + B 2 + G 2 = 1, 
be' — b'c ca' — c'a ab' — a'b 
V(i-0 2 )’ V(l-0 2 )’ V(i-0 2 )’ 
where 6, = cosine-inclination, = aa' + bb' + cc'. 
Hence, shortest distance — D — D' 
= A (a — a.') + B (/3 — /3') + G (y — 7') 
={( hc> - h ' c ) ( a - °0 + ( ca ' - c ' a ) (z 3 - &) + i ah ' - a ' b )1 
1 
V(l-0 2 ) 
l 
{a' (c/3 — 67) + b' (ay — ca) + c' (ba — a/3) 
+ a (c'yQ'- b'y') + b (ay - c'a) + c (b'a - a'ff)} 
— //i ¿te\( a f / + b 9'+ c h' + a f+ b '9+ c 'h), =S suppose. 
V(i - v) 
The six coordinates of the line of shortest distance are A, B, G, F, G, H, where 
A, B, G denote as before, and F, G, H are to be determined. 
Since the line meets each of the given lines, we have 
Af + Bg + Ch + Fa + Gb + He = 0, 
Af + Bg' + Gh! + Fa' + Gb' + He' = 0, 
and we have also 
FA+GB+HC= 0, 
which equations give F, G, H. Multiplying the first equation by b'G—c'B, the second 
by Be — Gb, and the third by be’ — b'c, we find 
(b'G-c'B) (Af+ Bg + Ch) + (Be - Cb) (Af + Bg' + Gh') + F 
Here 
a , b , c 
a', b', c' 
A, B, G 
= 0. 
b'G — c'B = 
V(1 -0>) 
1 
V(l -6*) 
l 
V(1 -0 2 ) 
[b' (ab' — a'b) — d (ca' — c'a)} 
[a (a' 2 + b' 2 + c' 2 ) — a' (aa' + bb' + cc')} 
(a — a'd),