68 
ON THE SIX COORDINATES OF A LINE. 
[435 
435] 
are said to be the ‘ six coordinates ’ of the line : as only the ratios of the six quantities 
are material, and as the last-mentioned equation establishes a single relation between 
these ratios, the system of the six coordinates contain four arbitrary ratios or parameters, 
for the determination of the particular line. 
formulae 
And I 
be fixe< 
x=0, y 
5. A line is thus determined by its six coordinates (a, b, c, f, g, h), which are 
such that of+ bg + ch = 0 ; and conversely any six quantities (a, b, c, f g, h) satisfying 
this relation may be taken to be the six coordinates of a line. 
of a lii 
values < 
to. An 
geometr 
6. It is proper to show that the ratios a : b : c : f : g : h are independent of 
the particular two points on the line, or two planes through the line, used for their 
determination. In fact, if instead of the points 
(a, b, c 
a, /3, y, 8, 
a, /3', 7', 8', 
9. 
we have any other two points on the line, say the points 
Xu -p get, X/3 -p g/3 , Ay -f- gy , \8 -P g8 , 
va+pu, v/3 + p@', vy + py\ v8 + p8', 
then if 
we hav 
&c. the 
then the six determinants have their original values each multiplied by Xp — gv; and 
the ratios are unaltered. 
And the like is the case, if instead of the planes 
A, B, G, D, 
A', B\ G', U, 
which 
be rem 
we have any other two planes through the line, say the planes 
the sysi 
A A + gA', A B + gB', A G + gG\ A D + gD', 
vA + pA', vB + pB', vG + pG', vD + pD' y 
10. 
the determinants have their original values each multiplied by Ap — gv \ and the ratios 
are unaltered. 
7. It may be remarked, that the theory of the six coordinates considered as derived 
from the two points (a, /3, 7, 8), (a', /3', 7', 8'), and as derived from the two planes 
(A, B, G, D), (A', B', C', D’\ is precisely the same in each case; and we may confine 
ourselves to the first point of view, regarding therefore the six coordinates as derived 
from the two points (a, ¡3, 7, 8), (a', /3', y', 8'). I further remark, that I do not at 
present in anywise fix the absolute magnitudes of the coordinates (a, b c, f, g, h): it 
is only the ratios that we are concerned with. 
are the 
equatioi 
of the ( 
8. The values of the ratios a : b : c : f : g : h of the six coordinates do how 
ever depend on the particular coordinate planes x = 0, y = 0, z = 0, w — 0, made use of 
for their determination; and in the sequel it will be necessary to investigate the 
which 
obvious 
system