On Systems of Rays. 19 
to uniform media: the formula gives, for example, the six following general expres- 
sions, which enable us to introduce the coefficients of the function v, of the second 
order, instead of those of Q, if it be thought desirable so to do, in many of the gene- 
ral equations of the present memoir, as the expressions contained in (#°) would 
enable us to introduce Q instead of v, in many of those of the First Supplement : 
ca 1 ov Ov Peo ) ; 
3° — ya U ay + Spe _ aos 3 
Qe S27, 02), Zn 
= =F ( eg hay" = rae one =) 3 
SOx yl ( 3 Sep 1s ov ov ) 
— = =O aay ; 
vw of3? da? oa B R (N°) 
ea _ 1 ( 2 OV ev ov OL ) f 
aay | v'G \ ! 3asG . SySu)” BGR), op 
&2Q 1 200 ov en ov 
Srou vv ( ov opoy ae dacp va oyea i éa® ) z 
ea I 2 Ov oy ov ov 
Suda | vee (=. Syoa 1 Say +,” Sap 7 BB ): 
To make more complete this theory of the coefficients of the function Q, which 
determines the nature of the final uniform or variable medium by the manner of its 
dependence on the seven variables « + v x y z y, and is supposed to have been so pre- 
pared that Q+1 is homogeneous of the first dimension relatively to «7 v, let us 
investigate the connexion of these coefficients of Q with those of the simpler though 
less symmetric function v, considered as depending on the six other variables o 7 x y 
2 x by the relation Q=0. For this purpose we are to combine the differentials of 
that relation with the conditions of homogeneity (B*) (C'*), and with the following 
other conditions of the same kind, which are only useful in variable media, 
Fa 
Ee 
2 
0a 
+ 
eQ 
epee: 
orox 
ea 
80 _ 80 
YSese . one? 
oOF veo) 
* Sady 1S ordy oe ouey Savy 
0) ie 
© Sadz 
eQ 
ca ca 
aa 
Oroz 
ria Sede S23 
In this manner we find, for the first order, 
(0°) 
éu éu éu U = 
sQ= No — F bo — or — Rae — ty — ae ax) (Ps) 
that is 
ie 5) OL Ble bv 82). 
so 8c? or a2” 
Soy lp" OF WOR ae thedy > Sale  eSqw8OK = «A Bo (Q’) 
ap Omen a or we Set ye Oy” 
VOL. XVII. 
O 
