On Systems of Rays. 65 
we may consider this differential dV =vds as becoming now a homogeneous function 
of do, dd, dp, of the first dimension, such that 
dvds _ 8dV Vda 8Vey WV dz WV 
Sdp  Sdp dx Op Ty dp oz dp op’ 
dvds odV SV & ok sy OV oz WV 
$40 —3d0 du 361 Sy 30° Sz 3090” 
é.vds _ odV Bog or mle oy OV &z mor 
odo on ox op ty 3p | 82 op op’ 
(X*) 
the symbol d referring still to motion along a ray. In like manner we may consider 
the initial differential element of /’, namely v'ds', as a homogeneous function of the 
first dimension of do’, d6’, dg’, and then we shall find that the partial differential coeffi- 
cients of the first order of this function, are equal respectively to 
ov OV ov 
> Bp! ean 50! ras ye. > 
we may therefore generalise the fundamental formula (4) as follows 
* = sat wads d.cds 
sV= — + 70 80 ayy op 
}.' f: 1, owds' ; ae v ‘ds! 
om Sdp! rar 57 ia oo! — ny oo! oe an = By. @®) 
And the auxiliary functions JY, 7, correspond to the TiN more general func- 
tions, 
8V a Van BO ae Oe 2 OK be 
maak Ria: +oE era -P+pe +05 39 + P35 +8 grt 8 ag + $ <5 
of which the first may be regarded as a function of 
a Sv ow 0, 
> 39° Se p ’ ° > ¢ aK 
and the second as a function z 
ay av yaya 
Spo? S07.-86.2), Bp eae! ? Ba a x 
It is easy also to establish the following general differential equations of a curved ray, 
ordinary or extraordinary, and the following general integrals analogous to and in- 
cluding those already assigned for rectangular and oblique co-ordinates, 
odV _odV odV See sd BRAS 
Bgamritisp ?) “SSa00 medd ado 3 
Z 
ae (2) 
ap! = const. ; 5 
VOL. XVII. S 
OV 
= const. ; no = const. 
