= 
On Systems of Rays. 81 
near ray-lines from the points of that section; that is, the projections of these near 
ray-lines on this plane, all pass through this centre of curvature. The two rectangu- 
lar planes of curvature, or principal diametral planes, of the guiding paraboloid, 
may therefore be called the planes of extreme projection ; under which view they 
were considered in the First Supplement, for the case of an uniform medium, and 
were proposed as a pair of natural co-ordinate planes passing through any given 
straight ray. The two planes of vergency, for the case of straight final rays, were 
also considered in that First Supplement, in connexion with the two developable pencils 
or ray-surfaces which pass through a given straight ray, and of which the two tangent 
planes contain rays infinitely near, and therefore coincide with the two planes of 
vergency. 
When the planes of vergency are real and distinct, then, whether the final rays are 
straight or curved, there exist two guiding lines perpendicular to the given final ray- 
line, which are both intersected by all the near final ray-lines from the points B, on 
the given final plane of wy, and which therefore suffice to determine the geometrical 
arrangement and relations of that system of final ray-lines. ‘To prove the existence 
and determine the positions of these two guiding lines, let us examine what conditions 
are necessary and sufficient, in order that a right line having for equations 
Y= 0 talie Ds, 2 — LZ, (Q") 
should be intersected by all the near final ray-lines of the system (A). These con- 
ditions are 
Z=r-+n cotan. ® = t—n tan. ©; (R") 
they give 
2 2n 
sin. 2@= sar (S") 
and 
(Z4 —r) (2-1) +m=0: CES) 
when therefore 
(t a ry >4n’, (U") 
that is, when there are two real vergencies there are also two real guiding lines of the 
kind explained above ; and these two guiding lines are contained in the two planes of 
vergency, and cross the final ray-line in the two corresponding points in which it is 
crossed by other ray-lines of the same system: the intersection of each guiding line with 
the given final ray-line being the point of convergence or divergence of the near ray- 
lines contained in that plane of vergency which contains the other guiding line. When 
the constant of deviation vanishes, these guiding lines are necessarily real, and are 
the axes of the two principal circles of curvature of the guiding paraboloid. And when 
the final rays are straight, then, whether 7 vanishes or not, the two guiding lines (if 
VOL, XVII. us 
