On Systems of Rays. 17 
if, as before, by virtue of (J’), we consider o, r, v, as functions, each, of x and of two 
other independent variables 0, , considering similarly o’, 7’, v', as functions, each, by 
(N'), of three independent variables 0,¢,x 3 and 7 as a function of the five inde- 
pendent variables 0, ¢, 6’, ¢> x: If we choose the independent variables 0, 4, so as to 
coincide with o, 7, and if in like manner we take o’, 7’, for the independent variables 
0, ¢, making, by (HZ), 
ov a ov’ i Ope 1 &’ : 
eA MRE Bay © 
and considering 7’ as a function of the five independent variables o, +, 0’, 7’, x, we 
have the following transformed equations for the extreme straight portions of a 
polygon ray, ordinary or extraordinary, 
_ = 7 . — B. oT 
O=u2 Pid A ha a Se . 
Oa ee. o= i B py oes ( ) 
=: y Se’? =y 7 @ BES 
which are analogous to the equations (J/*) and, like them, will often be found useful. 
It may be remarked here, that from the differential equations (0) of a curved ray, 
ordinary or extraordinary, to which, in the present and former numbers, we have 
been conducted by so many processes, the following may be deduced, 
(T*) 
dVv= dT=(«2+ yt 2) ds=0da + ydr +2du 
We may also remark, that when the final medium is uniform, and when therefore 
the quantities «, +, v, x, are connected by a relation (’), the quantity 
W (6 +r+v°)? 
may, in general, by means of this relation, be expressed as a function of 
oc T 
, , , 
=5 =o 2 z 
Re ae 9Ys2%5X> 
and that ZT’ (o+7°+ v7? may, in like manner, be expressed as a function of 
o Le , , ‘ 
ME O5T9VU9 X35 
and that therefore JV, 7, may both be made homogeneous functions, of any assumed 
dimension n, relatively to «, 7, v, so as to satisfy the following conditions 
W 
Be tae I 
Seite” gen SB . 
sr. OT Or (Us) 
t— tu— =n, 
Beedle to 
VOL. XVII. F 
