62 Mr Larmor, On the Form and Position [Jan. 31, 
correspond, so mark the point on the other retina that corre- 
sponds to each; these points determine two lines on the retinas 
which correspond to each other; the line in space constructed 
from them passes through the point considered. Thus through 
any point in space passes usually one, and only one, line of the 
system. 
Now it was shown by Sir W. R. Hamilton*, that the well- 
known propositions of Malus respecting the shortest distances of a 
normal to a surface from the consecutive normals, and the locus of 
their points of intersection, can all be extended to the general case 
of a system of rays which satisfy two conditions. One of these 
propositions is that each ray of the system is intersected at two 
points on its length by a consecutive ray, and therefore that each 
ray of the system is a bitangent to a focal surface which is the 
locus of these points, and is the analogue of the surface of centres 
in the simpler case of the normals. 
In the system under consideration, one, and only one, ray can 
usually be drawn through any point in space; therefore a point of 
intersection of two consecutive rays is a singular point, and must 
be the point of intersection of an infinite number of rays, forming 
acone. The focal surface, which is the locus of such points, must 
therefore degenerate into a curve. Further, through any point in 
space can be drawn one line which meets this curve twice; there- 
fore all conical projections of the curve possess only one double 
point; therefore the curve is a twisted cubic, and is the partial 
intersection of two quadric surfaces, but it may degenerate into two 
lines. 
It may be here remarked that the congruence of the first order 
of Pliicker is a more special form, corresponding to two linear 
relations between the six co-ordinates of the ray: for it the focal 
surface degenerates into two straight lines, each of which is met 
by all rays of the congruence. 
5. The results just proved, when modified by projection, give 
geometrical theorems of interest as being the extensions to space 
of three dimensions of well-known plane theories. Thus, if two 
homologous systems of planes pass through two given points, the 
lines of intersection of corresponding planes are the chords of a 
twisted cubic curve: if two homologous systems of rays pass 
through two fixed points, the points of intersection of those 
corresponding rays which intersect lie on a twisted cubic curve. 
Cases in which the cubic breaks up into a line and a plane conic 
are examined in detail by Helmholtz. 
* [Previously by Monge in 1781: see Prof. Cayley in Proc. Lond. Math. Soc. 
xiv. p. 139.] 
t+ Salmon’s Solid Geometry, Appendix m1.: Salmon, loc. cit. § 364. 
