On Systems of Rays. 55 
functions, having been sufficiently explained by the remarks made at the beginning of 
the seventh number, and by the details into which we have since entered ; we shall 
confine ourselves, in the remaining research of such connexions, for the new auxiliary 
function 7; to the case of extreme uniform media. And haying already treated of 
the mutual connexions between the coefficients of the two functions / and W, it will 
be sufficient now to connect the coefficients of either of these two, for example, the 
coefficients of W, with those of 7, of the first and second orders: since the connex- 
ions between the coefficients of all three functions will thus be sufficiently known. 
We shall also suppose that W has been made, before differentiation, homogeneous of 
the first dimension with respect to o, 7, v, that our results may be the more easily 
combined with the symmetric expressions already deduced from ‘this supposition, 
expressions which can be generalised in the manner that has been explained: and 
similarly we shall suppose that Z’ is made homogeneous of the first dimension with 
respect to o, 7, v, and also with respect too’, 7,v. Let us then seek to express the 
partial differential coefficients of the two first orders, of 7, by means of those of 
W, both functions being thus symmetrically prepared. 
In this inquiry, we have, as before, the conditions of homogeneity (U*) (/”*), 
relative to the function W, and analogous conditions relative to ZT, namely, for the 
first order, 
Mie oar 
Le ; Te + vy =F ; 
(V*) 
ol ,Or pols 
monk OR So Wane 
and, for the second order, 
dk oT eT er A Mi oT 
O=c55 Pr ey te OO tet eat Yas | 
PRONE Niet wet D nT wee BE 
Oadr or orev ” ~~ — 8a'dr’ or® or'du’ ” 
pee Se OC one ae aed ae 5 
Oadu orév oe” ~ — 8o'8u’ Or'dv’ ou’? ” 
Oe OE) oT oT OLE fOr Le Ole ,or A 
in’ ° dota’ +" Bede! * * ByBel } Bo ° Bode * * Babe *Y Boar? FOV 
RS. ap FT My. ROT gM ET 
or’ Badr’ t Oar i “Sousa ton Sroc'ie" abr Or’ 7 S780"? 
OF 1 ONeee er oT OLS Neel One noe & 
Su 7 BoB) Sede” Bue 2 By Bude! Bue” Be? 
RU uth ped ld oT OL ore OL OE 
[a yan ey iy Ose Mero eye? J 
