114 Professor Hamitton’s Third Supplement 
vergency, or the tangent plane to the developable pencil of straight rays, correspond- 
ing to the other or conjugate point of vergency on the given final ray. 
When the final medium is variable, the three first equations (#7'), namely, 
37 ww 
aT I ee Sr Paar out? 
are to be differentiated with respect to o, 7, v; and thus we obtain 
ow 3 ew ow 
ee ti a —— a dadu v0, 
SV. SW. SW. + 
we eee (Q 
ow ow ow 
Saou oo + Srdu aa due a 
and consequently, by elimination, 
ew ew ew, kW ew ew _ ew ey ew ea ewe Hy Re 
Sai BF Sul << Batr-Orlu bods Se Sey) Wa aba? 1 but Gear? = ae) 
this equation, therefore, (which may be put under other forms,) takes the place, when 
the final medium is variable, of the quadratic (Z") for a final uniform medium ; and 
if we eliminate from it « z v by (47), it will give, for any proposed initial point and 
colour, the equation of the single or multiple caustic surface, touched by the curved 
rays of the corresponding final system. 
The auxiliary function Z’ may also be employed for the case of curved rays, but it 
is chiefly useful when both the extreme media are uniform. In that case the extreme 
portions of a luminous path are straight, and we may employ for these extreme 
straight portions the equations (S*) under the form 
oS OSE, os ; oS ae 
=.= gos i Game 8 Sago. ane snl (S") 
in which we haye put, for abridgment, 
S= T-2zv+ev, (Ee) 
and in which we consider v as a function of o, 7, x3 v as a function of o’,7, ¥; Tas 
a function of o, 7, 0,7, <3 and S as a function of 2, 2’, o, 7, o', 7, x. Differen- 
tiating these equations (S") with respect to o, 7, o', r', we find that if the extreme 
straight portions, ordinary or extraordinary, of two infinitely near paths of light of 
the same colour, intersect in an initial point 2’ y' 2', and in a final point x y 2, the 
final and initial variations 86, $r, 8’, 8r', and the final and initial ordinates of inter- 
section 2, 2’, must satisfy the four following conditions, 
