88 
ON THE SIX COORDINATES OF A LINE. 
[435 
56. Hence writing 
(X, Y, Z, W) = 
the foregoing equation is 
-h, 
9> 
- a, 
K - g, 
• , /, 
-/> • > 
-b, - c, 
a, 
b, 
c, 
(x, y, z, w), 
bc.wX + ca.wY + ab.wZ — gh.xW — hf.yW —fg.zW 
— af{wW + xX) —bg (wW + yY) — ch (w W + zZ) 
— bf.yX —cg.zY —ah.xZ —cf.zX —ag.xY —bh.yZ = 0\ 
or, collecting and arranging, this is 
X {—af. x — bf. y — cf. z + bc . w\ 
+ Y {—ag.x — bg.y — cg.z + ca.w} 
+ Z {—ah . x — bh . y — ch . z + ab. w) 
4- W {-gh.x -hf.y -fg . z + {af. + bg.+ ch .) w} = 0, 
which is satisfied by X = 0, F=0, Z= 0, W=0; that is, since (a, b, c, f g, h) have 
been written in place of (a lf b lt c x , f, g x , K), by X x = 0, ^ = 0, Z x — 0, 1^ = 0 (if we 
thus denote the corresponding functions of (a x , b x , c ly f, g u hf), that is, the hyperboloid 
passes through the line (a x , b 1} c lf f it g u and similarly it passes through the other 
two lines. 
Article Nos. 57 and 58. The six coordinates defined as to their absolute magnitudes. 
57. In all that precedes, the absolute magnitudes of the coordinates have been 
left indeterminate, only the ratios being attended to. But the magnitudes of the six 
coordinates may be fixed in a very simple manner as follows; viz. using ordinary 
rectangular coordinates, then for any line, if x 0 , y 0 , z 0 are the coordinates of a particular 
point on this line, and a, /3, 7 the inclinations of the line to the axes, the coordinates 
of another point on the line are 
x 0 + r cos a, y 0 + r cos /3, z 0 + r cos 7; 
and hence writing 
# 0 + rcosa, y 0 + r cos ¡3, ^ 0 + ^cos7, 1, 
^0 > y<> > ¿0 1 I) 
we have 
a : b : c : f: g : h = z 0 cos /3 — y 0 cos 7 : x 0 cos y—z 0 cos a : y 0 cos /3 — x 0 cos a : cos a : cos /3 : cos 7. 
Or we may take 
a = z 0 cos /3 — y 0 cos 7, f— cos a, 
b = xo cos 7—^0 cos a > g — cos & 
c = y 0 cos a — x 0 cos /3, h = cos 7,