On Systems of Rays. 13 
of which indeed the first is general. But if the final medium be uniform, then 
W remains an arbitrary function of the four variables o, 7, v, y, which are in this 
case connected with each other by the relation (A); and the two equations (D’) 
(K’), together with the two first of those marked (J'), compose a system, which is a 
form for the integral of the partial differential equation 
OV OV BV 
ae? oy 5) Oe ’ x) 
to which the first equation (C) in this case reduces itself. In like manner, if both 
the extreme media be uniform, in which case the second equation (C’) reduces itself 
0=2 ( (V) 
to the form 
Sega eS 
the system of the partial a equations (7) (JV) has for integral the system 
composed of the equations (F”) (4’) (V’) (0'), and the two first equations (JZ’), in 
which 7’ is considered an arbitrary function of «, 7, v, o, 7, v, x. It will be found 
that these integrals are extensively useful, in the study of optical combinations. 
The two partial differential equations, (/”’) (/V’’), of the first order, are them- 
selves integrals of the two following, of the second order, 
SV ET BY BV EV BY _ 
- dy? 82% | Sady Sydz dzdxr 
2 SV (&V SV (eV > 
or ae ‘ti oy? Or ae (X) 
and 
SV SV SV SV SV &V 
ae by? B22 T ws ay Bye B78e' 
eV % yr 
au? Ce oF aa) seen mo @) 
which are obtained by elimination from (@Q) and (X’), after making 
ee 0, oo =i; a = (0); 
ox oy oz 7 
OH Be Cae ( ) 
The system of the three first of these six equations (Z’), or the partial differential 
equation of the second order (X’), or its integral of the first order (V’"), expresses 
that the final medium is uniform ; and the uniformity of the initial medium is, in like 
manner, expressed by the three last equations (Z’), or by the partial differential equa- 
tion ( Y"), or by its integral of the first order (/”’).. The integral systems of equa- 
tions, also, which we have already assigned, express properties peculiar to optical 
combinations that have one or both of the extreme media uniform. 
VOL. XVII. E 
