DE. PLUCKER ON A NEW GEOMETRY OF SPACE. 
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II. Complexes. Congruencies. Surfaces generated by a moving right line. Developable 
surfaces and curves of double curvature. 
11. A homogeneous equation between any six line-coordinates is said to represent the 
complex of those lines the coordinates of which verify that equation. According to the 
identity of ratios I. to VIII., the following equations, 
F [{uv'—u'v), —{tv'—t'v), (tu'—t'u), {tw'—t’w), {uw'—u'w), {vw'—v'wj]= 0, 
F[(xvr'— x 1 ™), {y^ 1 —y'm), {zzj' — z'vs), {yz'—y'z), —{xz'—x'z), (xy'—x'yj] = 0, 
F[cosa, cos/3, cosy, cicosX, ^cos^, &cost'] = 0, 
F [fudv—vdu), —{tdv—vdt), { tdu—udt ), dt, du, dv]= 0, 
F [dr, dy, dz, {ydz — zdy ), — {xdz — zdx), {xdy—ydx)~\= 0, 
F[X, Y, Z, L, M, N]=0, 
F [r, s, 1, (— «r), §, >?]= 0, 
F[(— *)» ^p, q , 1]=0, 
represent the same complex ; F being supposed to indicate always the same homogeneous 
function of the different groups of line-coordinates. The complex is said to be of the nth. 
degree , and represented by if its equations are of that degree. 
12. Starting from the first equation, 
Q re =F[(W— u'v), —{tv'—t'v), {tu'—t'u), {tw'—t'w), {uw'—u'w), {vvJ —v'w)~\= 0, . (1) 
t, u, v, w and t', u', v', w' are to be referred to any two planes passing through any line 
of the complex. Let one of the two planes {t 1 , u', v', w') be any given one. Then the 
last equation, in regarding t', u', v', w' as constant and t, u, v, w as variable, represents 
within the given plane a curve enveloped by tangent-planes {t, u, v, w). The lines of 
the complex, confined within the plane, also envelope the same curve, the class of which 
is the same as the degree of the complex. Hence 
A complex of the nth degree being given, in each plane traversing space there is a 
curve of the nth class enveloped by lines of the complex. 
The equations of such curves fully agree with the general equation of the complex 
itself. We have only to consider in this equation t' , u', v' , w' as constant in referring 
them to the given plane, while t, u, v, w are regarded as variable plane-coordinates. 
If %=1, the curve in each plane is replaced by a point; each line within the plane 
passing through that point belongs to the linear complex. 
If n= 2, the curves enveloped are conics, which may degenerate into systems of two 
real or imaginary points. 
13. If, in the second equation of the same complex, 
xn n =F[{x-x'), {y-y'), (: z-z '), {yz'-y'z), -{xz'-x'z), {xy'-Ay)~\— 0, . (2) 
where we put &'=■&=. 1, and X denotes a constant, x', y' , z' are referred to any given 
