On Systems of Rays. | 
If we differentiate the first equation (C) relatively to 2’, 7’, 2’, we find, by the fore- 
going number, 
Dy aaah eV 
ye “Ovba" +.B ier bye’ + Y Sede? 
Freep SV eV 
° Bray 7 B Syay’ | Y Sedy'’ (U) 
0 OV eV Vv 
mt OL 
scar + Bayar + Y gar? 
of which, in virtue of (D), any two include the third, and which may be put by (P) 
under the form 
V V 
oad <>; 0nd ;0=d 55 5 (V) 
and these differential equations (7) of the first order, in which the initial co-ordi- 
nates and the colour are constant, belong to the ray, and may be regarded as integrals 
of (OQ). They have, themselves, for integrals, 
a= const. , = 4 a = const. , (W) 
the constants being, by (B), the values of the initial quantities 
i By 
a> 8B * ky’ | 
In like manner, by differentiating the last equation (C’), we find the following 
equations, which are analogous to (@Q) and (UV), 
= const. 
we Po Der ae aN 
: aad 77 ale 97 em 
Sr SES Oi NB 
a suey +O gat leayatin aay (x) 
, 3V BV a 
* 3yo7 * Payer FY oom a BA? 
ef ala. gr ee Ne 
* Boy = ce a 
and 
mmowea 
- ia + Bs by ue a 
37 
Y 
rn oe Pay” Y Syazt i (®) 
Axe a pale 
952 sae iPS s +7 Seg 
The second members of the three first equations (X) vanish when the initial medium 
is uniform, and those of the three first equations (@) when the final medium is so ; 
and in this latter case, of a final uniform medium, the final portion of the ray is 
