1887.] of the Horopter. 63 
6. The curves and lines hitherto considered are seen singly 
because their images occupy corresponding lines on the retinas; 
but it is not necessary for this that the images of a definite point 
on one of them should occupy corresponding points. This would 
be a more difficult condition to fulfil, and only holds for the points 
of intersection of lines of the group, for then the image points are 
determined as lying on each of two lines. It is in fact clear that 
through such a point a singly infinite series of lines of the group 
can be drawn, forming a cone: for through its images on the retinas 
a singly infinite number of corresponding pairs of lines can be drawn, 
and each pair determines one of the group*. Thus the locus of points 
seen singly is the twisted cubic curve which is the nudal curve of 
the congruence. 
It is worth while to point out that the pairs of points on one of 
the lines of the congruence whose images occupy corresponding posi- 
tions on the retinas form a geometrical involution; that the line 
~ meets the horopter curve in its double points ; and that its foci are 
therefore equidistant from these points. 
7. Ifa third condition is given between the parameters of the 
line, the locus of the line becomes a ruled surface; if this new 
condition is linear, e.g. if the images of the line on the retinas 
correspond under normal conditions to a horizontal line or to a 
vertical line in space, the surface is a ruled hyperboloid. 
These two surfaces are the horizontal and vertical line horopters 
of Helmholtz, and the cubic curve is the point horopter. 
8. Inasmuch as the field of simultaneous vision of the eye is 
necessarily small, the most important part of the point horopter is 
that im the neighbourhood of the point on which the eyes are fixed. 
And it may be observed also, that the more oblique portions of the 
field of view do not practically come under the conditions of the 
geometrical problem, for the image on the retina cannot be con- 
sidered as plane, except in its central portions. 
The horopter may therefore be identified with the tangent to 
it at that point for most practical purposes. It seems therefore of 
importance to obtain the direction of this tangent line, especially 
as the results come out to be comparatively simple. At all points 
in the field of view in the neighbourhood of this line, the binocular 
vision will preserve completely the single character. 
The result obtained will apply immediately to all the simpler 
cases in which the horopter curve breaks up into a line and a conic, 
the only ones that have been completely discussed+. For the 
more general case the angle of rotation @ of this section is given 
* Helmholtz, Wissen. Abhandl. 11. p. 488; Physiological Optics, § 31. 
+ Helmholtz, Physiological Optics, § 31; Hermann, Physiologic, 7 ed. pp. 419-25, 
WOES Wile Tee IM ; o 
