On Systems of Rays. 63 
8V 8V 
Saf = const. ; ay const. ; ier const. : (O*) 
while, if the final portion of the ray be straight, we have also, for that final portion, 
av av Bul ; 
Salat const. ; ae const. ; ate const. CE) 
The formula (4’) of reflexion or refraction, ordinary or extraordinary, namely, 
AV=0, 
extends to oblique co-ordinates ; and if we introduce new auxiliary functions W,, 7,, 
analogous to /V, 7, and defined by the new equations 
Via = Vite, OY) TZ, Ui, 
(Q") 
analogous to the definitions (D’) (£"), and attend to the meanings and properties of 
the symbols o, 7, v, ¢/ z/ v/, we shall obtain the following expressions for the varia- 
tions of VY, W, T., 
V 
SUL w= o On, — o/ 8x; 35 TOY, = 7/éy, + voz, —v/dz; 55 = ox 5 ] 
‘ rd ‘ ‘ fd Ve 
SW = 480, +n! +y Sr, +7'8y' +2804 0/82! — = ay; (R') 
= (ee ne res 
f= =W -2, oF WY, a, FES; Us 
3 qs = 2 80,— x8; +Y or, —y, or, =f zou, = Zev; = = ox 3 
which resemble the expressions (4‘) (B’) (C’), and lead to analogous results. Thus, 
the partial differential coefficients of the new auxiliary functions JY7,, T,, may be 
deduced, by methods similar to those already employed, from the new coefficients of 
the characteristic function V, which may themselves be deduced from the old coeffi- 
cients of that function, by the following general formula, 
QE Oe Oe) 
( Xe, = + Ys, Ln Za! a ( tees + Yr + 2! =) 
(S*) 
( Ly, + Yy, = + %, =)" ( ays a + yy! + + Zy! i 
( Fy = + Ys, 3, + 2 =) ( at 5? ~ - ies + 2a 
and the equations of a straight final ray may be put under the forms, 
1 8wW,. 1 ANE OW, 
hie ED AB. (y- Sr, =; (, ~ Sy, ) ic 
Loewe ears a ee ar oe 
a, (2- os, ap, @- or, )=— (, a ou, )» 
while those of a straight initial ray may be put under these other forms, 
