782 DE. PLtJCKEE ON A NEW GEOMETET OF SPACE. 
equation, in regarding the coordinates as variable, represents in ordinary point-coordi- 
nates the surface which in line-coordinates is represented by the system of the three 
equations (5). 
18. Likewise there are in each plane traversing space three curves enveloped by lines 
of the three complexes Q ! , O", QI". In the general case these curves have no common 
tangent. In certain positions of the plane they have, and then the common tangent 
belongs to the surface (O', O", O'"). Reciprocally, within a plane passing through any 
generating line of the surface, the curves enveloped by the lines of any complex O touch 
the generating line, and continue to do so if the plane revolves round it. The plane in 
each of its positions is a tangent-jplane of the surface. 
Put 
O' = F [(uv'—u'v), —(tv'—t'v), (tu'—t'u), (t—f), (u—u 1 ), (v— P)] = 0, j 
0" = F"[(W— u'v), —(tv'—t'v), (tu'—t'u), (t—f), (u—u'), (v— P)]=0, l • (9) 
0"'= F" \_(uv' — ulv), —(tv'—t'v), (tul—t'u), (t—f), (u—u'), (v— P)]=0. ] 
In regarding t, u, v as variable plane-coordinates, and referring t', u', v' to the tra- 
versing plane, these equations represent, within that plane, the three curves enveloped 
by lines of the three complexes O', O", O'". On this account they may he reduced to 
equations between two variables only, and therefore will not, in the general case, be 
verified by any values of the three variables reduced to two. By eliminating the 
variables between the last three equations, an equation, 
^(f, u!,v')= 0, (10) 
will be obtained, which, if t', u', v' are regarded as variable, represents in plane-coordi- 
nates the surface (O', O", O'"). 
19. In order to derive the equations (9) from the equations (6) (both systems of equa- 
tions representing the same surface), we may first pass from (6) to the three new equa- 
tions, 
F W-fz), -(xz'-x'z), (xy'-x'y), (x-x'), (y-y'), (z—z')]=0, 
F '[(yz'-y'z), -(xz'-x'z), (xy'-x'y), (x-x'), (y-y 1 ), (z-z’)]=0, 
Y"\(yz'-y’z), -(xz'-x'z), (xy'-x'y), (x-x'), (y-y'), (*-*')]= 0, 
and then replace x, y, z, x', y', z' by t, u, v, f, u’, v'. The last equations are likewise 
obtained by merely exchanging amongst themselves the constant coefficients in each of 
the three equations (6). The way of exchanging is obvious. Hence, in considering 
that the equation (10) is derived exactly by the same algebraical operations from (9) as 
(8) from (7), we may conclude that (10) may be derived from (8) by a mere exchange of 
constants and a substitution of plane- for point-coordinates. 
20. In a congruency (O n , Q m ) there are mn lines meeting in a given point. Two, 
three, four of these lines may coincide. In this case the cones of both complexes 
Q n and O m , the common centre of which is the given point, are tangent one to another, 
or osculate each other along the double or multiple line. In order to get the analy- 
