On Systems of Rays. 43 
Hence may be deduced, by reasonings analogous to those already employed, the fol- 
lowing formula for ©’ JV, which is equivalent to twenty-eight separate expressions for 
the partial differential coefficients of W, of the second order, considered as deduced 
from the coefficients of 7’, on the foregoing suppositions of homogeneity : 
ate ll ee 2 ae (T— WSO + 1730 4 8W (8,0! ~80) + 23,177.80 
ow" 
* Ge a) D+ 8 (ons Sy; Fen) D'D! 
a sr)" +2(R5; -W T2,)D'D 
Ct. = "aC; = WV ae )DD ; (R’) 
in which we have put for ratte 
p-* G (av +20043 2 Wao 7) — 20" ( ay’ +980 43, =), 
D=% 
O (ar +0100 +352 — wa) 88 ( ary sara _ ws, my (s’) 
By 
ra Oy ty a) 3a’ Bai acon Ban 
Ais +9304 3,27 7 — We) = << ( ar +a'80+8 5 - Was), 
and in which W" can be deduced from 7} by the relation 
o?47' +0" ee we) (= = we) & ‘a = ee 
vw” — \de8 Sait \B72 372 Sadr’ Sale’ 
er ea! or fa! oer 820! \2 
seem asso arate Monee «) i; Gael areas 
er S0'\ (327 80! er So! \? . 
+ (ge - Waa) Ge Wie) - (ag - Wines) 
General Remarks and Cautions, with respect to the foregoing deductions. Case of 
a Single Uniform Medium. Connexions between the Coefficients of the Function 
v, Q, v, for any Single Medium. 
10. We are then able, by combining the formule of the three preceding numbers, 
to deduce the partial differential coefficients of the two first orders, of any one of the 
three functions 7, W, 7, from those of either of the other two, when the extreme 
media are uniform and known: since we have expressed the coefficients of Y by 
those of JV, and the coefficients of W by those of 7, and reciprocally, for this case 
