18 Professor Hamiiton’s Third Supplement 
With this preparation, the two first equations (J’), and the two first equations (mM"), 
which belong to the straight final portion of the ray, may be transformed by (1) to 
the following, 
6Q oT oQ 
x Tiss (ox +1y +02)=2—n nS - 3o —aTe s 
6Q a éT 6Q 
y “ee (ox +1 +w2)=o— nW = Slag nT — , (V*) 
Ss, = (ow + ry tve)ae—n ee _— nT. 
If then we make »=1, that is if we make JV homogeneous of the first dimension 
relatively to o, 7, v, and if we attend to the relation (D’), we see that the equations 
of this straight final portion may be thus written, 
a W _ ow 
t= +V ay =D oF yee 
—& or 
of which any two include the third, and which we shall often hereafter employ, on 
account of their symmetry. 
In like manner, when the initial medium is uniform, and therefore the initial por- 
tion straight, the equations ( 0’) of this straight portion may be put under the form, 
Lee ye = (We?) 
; iy ai i Fj 
Xu =e (o2' +ry oe ene Bes 
; “ er pit dtc gee a ; 
y¥-3 (ov try tuz)= — Se +1To, (&) 
ae ae ee oT 1p! 
2 a (ca +ry t+uz)= a +uT >_> 
by making 7 homogeneous of dimension 7’ relatively to o', r, v, so as to have 
of or + or +uv , oT 
Cat bv’ 
If both the extreme media be uniform, and if we make 7=0, m =O, that is if we 
express /V as a function of 
=<0T. (X) 
o T t ' 1 
mY aL U,Y,2;, X 
and 7’ as a function of 
, , 
r 
Cy ear, ara © 
Oe OMe x 
we find the following forms for the equations of the extreme straight portions of a 
a polygon ray, ordinary or extraordinary, less simple than (S*), but more sym- 
metric, 
