16 Professor Hamitton’s Third Supplement 
independent variables 6, $5 x» x’, y', 2. We may choose o,7, for the independent 
variables 0, , considering v as, by (C’), a function of o, 7, x, such that by (#7), 
du a ov B 8 _1 & 
Se Se eg i L’ 
oo y? & vy’ 8x 8x’ ry) 
and considering JV as a function of the six independent variables o, r, x, 2’, y', 2’ 5 
and then the equations (J*) or (*), for the final straight portion of a polygon ray, 
ordinary or extraordinary, will take these simpler forms, which we shall have frequent 
occasion to employ, 
ow 
or * 
The other auxiliary function, Z, gives the following equations between «, r, v, for 
the final portion, straight or curved, when the initial medium is variable, 
(M’) 
pte NG ee oe 
oY, i 
dH oT 
—— = const., yr = const. z =const., (N?) 
So > Ru" 
in which o’, 7’, v’, belong to some point on the initial portion, and in which the con- 
stants are, by (L’), the negatives of the co-ordinates of that point; it gives, in like 
manner, for the initial portion, when the final medium is variable, the following equa- 
tions between o’, 7’, v’, 
oT 
oT 8T A 
= const., 3 = const., > = const., (O°) 
cs, t, v, belonging to some point upon the final portion, and the constants being the 
co-ordinates of that pomt: and from these equations we might deduce the differential 
equations (QO), by processes analogous to those already mentioned. When both the 
extreme media are uniform, and therefore both the extreme portions straight, we 
have, for these straight portions, the following equations, deduced from (1/’) ( 0‘) (7), 
1 or 1 oT I oT 
5 alii gn) = Ca 
1 hab ho ih 1 ; Od 1 Petey 
“(2 + J=ply +s a7 Piso 
which may be thus transformed, 
(P) 
2 he ee a SY 
60 CO” ~80 86 80” 
eS a > aa 
op op 8p Op” ; 
028 Sree ene SE ne 
oy’ 3 36 | 8H’ 
ona Hy yy gO 
oY" oe by’ 8h"? 
