On Systems of Rays. 15 
ow ow ow 
OF ah ae aed 
ov. SW, SW 
i a (F) 
WwW 
jee Pe a gargs. 
dadz" ordz’ Ovdz 
d still referring to motion along a ray : and if we combine these with the following, 
wVew wesw kv ew 
Se Bode! * By Bria” * 82 Bde’? 
wesw wesw wv sew 
O= Be Say! * ay Bray’ + Be Bay"? 
we wWeW ww 
= Se Sole’ * dy Srdz’ + Sz Bude’ ’ 
which are obtained by differentiating the first equation (7"’) relatively to the initial 
co-ordinates 2’ y' z’, and by attending to the relations (K ), we see that for a curved 
ray the differentials do, dr, dv, are proportional to 
dv du bu 
we > ay 5) sz ? 
and from this proportionality, combined with the relation 
o= 
(G*) 
ads + Pdr + ydv=(a ant Bet 7) as, (H®) 
which results from (7) and (P), we can easily infer the equations (0): these differ- 
ential equations (Q) for the final portion of a curved ray, which can be extended to 
the initial portion by merely accenting the symbols, may therefore be deduced from 
the consideration of the auxiliary function JV. The equations (0) for a curved ray, 
may also be deduced from the function VY, by combining the differentials d of the 
three first equations (#7'), with the partial differentials of the first equation ( 7”), 
taken with respect to o, 7, v. 
The same auxiliary function /V gives for the final straight portion of a polygon 
ray, the two first equations (I’), which may be thus written, 
Bec geal 3 aw 
a 3 ae er eo) 
these equations may also be put under the form 
if in virtue of (JX’), we consider o, 7, v, as functions, each, of y, and of two other 
independent variables denoted by 0, ¢, and consider /V as a function of the six 
