On Systems of Rays. 73 
is or is not integrable ; and if there be any one surface perpendicular to all the final 
rays, there is also a series of such surfaces, represented by the integral of this equa- 
tion. Hence, in the present question, the normal surface sought is such, if it exist 
at all, as to satisfy the conditions 6z=0, and 
oz + dada + dBdy = 0 ; CX?) 
that is, if it exist, it must touch the given final plane of ay, and must have contact of 
the second order with the following paraboloid, which may therefore in our present 
order of approximation be employed instead of it, 
oa 2 éa 6B . 6p 2 — Th 
an +(S+e ley ts, 7 =0- (2) 
The normals to this paraboloid, near its summit, that is, near the final point of the 
given luminous path, or the origin of the final co-ordinates, have for their approx- 
imate equations, 
Qe4+ 
8a éa 
= br +2(> ox +5 ¥) + 2ney, “ 
y= +2(3 ox +3 ay ) —Znen , 
if we put for abridgment 
ab Gh ot», 10 
n=4 (5 -5)3 (B") 
they coincide therefore with the ray-lines (/”*) when the following condition is satis- 
fied, 
Joes 
oc by’ 
which is in fact the condition of integrability of the differential equation (X°), because 
we have made a B vanish by our choice of the axis of z. The condition (CC) is 
satisfied, by (#°), when the final medium is ordinary ; and in fact the final rays:whe- 
ther straight or curved are then perpendicular to the series of surfaces represented by 
the equation 
(C*) 
V =const. : ¢D*) 
which is, for ordinary rays, the integral of the equation (X°), and gives, as an 
approximate equation of the normal surface at the origin, the following, 
Sl eV i a 
Ogee 120! Uouztae Ge ta Yt hie Te (E") 
agreeing, by (#”), with the equation of the paraboloid (Z°). In general, the condi- 
tion (C') for the existence of a normal surface, may be put, by (@*), under the form 
VOL. XVII. U 
