DE. PLUCKER ON A NEW GEOMETRY OF SPACE. 
745 
On the supposition of rectangular axes of coordinates, the last equations become 
r= 
sin ct—r cos a 
cos u + r sin a 
cos a + r sin a 
cos a + r sin ci 
(sg — go - ) sin a — <r cos a 
cos a + r sin a 
(41) 
,, , , (so — r<r) cos a — o- sin i 
£> — rff =— : • 
cos a + r sin a 
«§ 
g — § 
. . . . (42) 
• • • • (43) 
In order to pass from the first system of coordinates to the second, r, s, g, <r and 
r\ s', g>', d are to be replaced by one another, while the sign of a is to be changed. Thus 
we get the following formulae : — 
sin ci -f- r 1 cos a 
i — 7 ’ 
cos a — r sin a 
§ = 
a = 
cos ci — r sin a 
V 
cos ci — r sin ci 
(s' g' — r'a') sin « + c' cos « 
cos u—r 1 sin « 
(44) 
( s'p 1 — r l <r l ) cos a — 
So — TG— i-a L 
b cos « — r sm 
<r sin a 
a 
(45) 
39. The general equation of the linear complex 
Ar+Bs+C + D<r+Eg> + F(sf — r<r)=0 . . (7) 
becomes, if the origin is moved to any point (x°, y°, z°) . . . (30), 
(A — Yy° — Ez°)r + (B + Fx° — D^°)s + (C + E#° + Dz°) -f Do-' + E^' + F(so — r<r) = 0. 
If 
D E — F 
the primitive equation is not altered. Consequently the complex remains the same if 
it be moved parallel to itself along a direction indicated by the last equations. We 
obtain in denoting by g, tj, £, the angles which this direction makes with OX, OY, OZ, 
COS 0 COS 7) COS £ 
~TT E T' 
(46) 
40. In order to get OZ congruent with a right line OM of the determined direction 
and passing through O, we may in the first instance turn the system of coordinates 
