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DE. PLUCKER ON A NEW GEOMETRY OE SPACE. 
obtained, which, combined with these, indicates that there is a certain number of points 
into which quadruple lines of the congruency converge. 
In congruencies of a peculiar description only we meet quintuple lines. 
21. In quite the same manner we may determine the position of planes within 
which two, three, four of the mn lines of the congruency (Q„, O m ) coincide. In that 
case both curves within the plane, enveloped by lines of the complexes Q, n and Q m , 
touch or osculate one another on a common tangent. 
In operating on the first two equations (9) as we did on the first two equations (6), 
we get, in order to represent in plane-coordinates the locus enveloped by planes con- 
fining a double line of the congruency, the following equation, 
\p(t, u, v)= 0, (16) 
which, as the remarks of No. 19 here likewise hold, is derived by a mere exchange of 
constants from (10). Each plane passing through a double line being an enveloping 
tangent plane of the represented surface, this surface degenerates into a curve of 
double curvature. 
Another equation may be derived from (15) in the same way. Let it be 
vj /(#, u, v)=0, (17) 
the system of the two equations (16) and (17) representing a developable surface , the 
tangent planes of which confine the triple lines of the congruency. Finally, there are 
certain tangent planes of the developable surface which confine the quadruple lines of 
the congruency. These planes, as well as the points of the curve of double curvature 
through which the quadruple lines pass, are determined by associated plane- and point- 
coordinates, both being functions of the constants of the congruency, and are obtained 
one from another by the above-mentioned exchange of these constants. 
22. The double lines of a congruency constitute a surface , degenerated into a deve- 
lopable one, as they envelope a surface, degenerated into a curve of double curvature. 
The developable surface is represented in point-coordinates by a single equation (13), in 
plane-coordinates by the system of two equations (16) and (17). The curve of double 
curvature is represented in plane-coordinates by a single equation (16), in point-coordi- 
nates by the system of two equations (13) and (15). The tangent-planes of the surface , 
confining triple lines of the congruency, osculate the curve ; the points of the curve , 
through which these triple lines pass, are osculating points of the surface , in which 
three consecutive tangent planes meet. The curve , in certain points where the tangent 
is an osculating one, is osculated by a plane in four points. Through such a point pass 
four consecutive tangent planes of the surface , the common intersection of which is a 
line of inflexion of the developable surface . The quadruple lines of the congruency 
pass through such points, and are confined within such planes*. 
* In two remarkable papers “ On a New Analytical Representation of Curves in Space,” published in the 
third and fifth volume of the Quarterly Journal of Mathematics, Professor Cayley employed before me, in order 
to represent cones, the six coordinates of a right line, depending upon any two of its points. Having lately 
