A SURFACE OF THE SECOND ORDER INTO ITSELF. 
Ill 
122] 
one point of the given pair, and the point the corresponding point to which has to 
be determined, take through each of these points a generating line, and take also 
two generating lines out of the given system of four lines, the four generating lines 
in question being all of them of the same set, these four generating lines inter 
secting either of the other two generating lines of the given system of four lines in 
four points. Imagine the same thing done with the other point of the given pair 
and the required point, we should have another system of four points (two of them 
of course identical with two of the points of the first-mentioned system of four points); 
these two systems must have their anharmonic ratios the same, a condition which 
enables the determination of the generating line in question through the required 
point: the other generating line through the required point is of course determined 
in the same manner, and thus the required point (i.e. the point corresponding to any 
point of the surface taken at pleasure) is determined by means of the two generating 
lines through such required point. 
It is of course to be understood that the points of each pair belong to two 
distinct systems, and that the point belonging to the one system is not to be con 
founded or interchanged with the point belonging to the other system. Consider, now, 
a point of the surface, and the line joining such point with its corresponding point, 
but let the corresponding point itself be altogether dropped out of view. There are 
two directions in which we may pass along the surface to a consecutive point, in 
such manner that the line belonging to the point in question may be intersected by 
the line belonging to the consecutive point. We have thus upon the surface two 
series of curves, such that a curve of each series passes through a point chosen at 
pleasure on the surface. The lines belonging to the curves of the one series generate 
a series of developables, the edges of regression of which lie on one of the surfaces 
intersecting the surface of the second order in the four given lines; the lines belonging 
to the curves of the other series generate a series of developables, the edges of 
regression of which lie on the other of the surfaces intersecting the surface in the 
four given lines; the general nature of the system may be understood by considering 
the system of normals of a surface of the second order. Consider, now, the surface 
of the second order as given, and also the two surfaces of the second order inter 
secting it in the same four lines ; from any point of the surface we may draw to 
the auxiliary surfaces four different tangents; but selecting any one of these, and 
considering the other point in which it intersects the surface as the point corre 
sponding to the first-mentioned point, we may, as above, construct the entire system 
of corresponding points, and then the line joining any two corresponding points will 
be a tangent to the two auxiliary surfaces; the system of tangents so obtained may 
be called a system of congruent tangents. Now if we take upon the surface three 
points such that the first and second are corresponding points, and that the second 
and third are corresponding points, then it is obvious that the third and first are 
corresponding points;—observe that the two auxiliary surfaces for expressing the corre 
spondence between the first and second point, those for the second and third point, 
and those for the third and first point, meet the surface, the two auxiliary surfaces 
of each pair in the same four lines, but that these systems of four lines are different