On Systems of Rays. 25 
Deduction of the Coefficients of WV from those of V. Homogeneous Trans- 
formations. 
8. Reciprocally, if it be required to deduce the partial differential coefficients of 
IV, of the second order, from those of /”, in the case of a final variable medium, 
we have only to compare the expressions for 
rr 
v7 / 
dx, dy, dz, 8a, dr, dv, —S—, 
as linear functions of 8c, dr, dv, da’, dy’, dz’, dx, deduced from the equations (4°), with 
those that are obtained by differentiating the seven equations (G@') (#'), into which 
(B) resolves itself: that is with the developed expressions for the variations of 
aw aw 3W 3 aw aw aw 
So” Sr ? by.” b2'’ BY ” 82” Sy 
But if the final medium be uniform, then (B’) no longer furnishes the seven equa- 
tions (G) (H’), nor can da, dy, 6z, themselves, but only certain combinations of 
them, be deduced from (4*); and the auxiliary function JV” is no longer completely 
determined in form, by the mere knowledge of the form of the characteristic function 
PV’, with which it is connected ; because, in this case, the seven variables on which 
W depends, are not independent of each other, four of them being connected by 
the relation (/X"), by means of which relation the dependence of JV” on the seven 
may be changed in an infinite variety of ways, while the dependence of /” on its 
seven variables, and the properties of the optical combination, remain unaltered. 
Accordingly this indeterminateness of JV”, as deduced from /’, in the case of a final 
uniform medium, produces an indeterminateness, in the same case, in the partial 
differential coefficients of 7 ; and whereas V7, considered as a function of seven 
variables, has thirty-five partial differential coefficients of the first and second orders, 
we have only twenty-seven relations between these thirty-five coefficients, unless we 
make some particular supposition respecting the form of JV; such as the supposition, 
already mentioned, that one of the related variables, for example v, has been removed 
by a previous elimination, which gives the eight conditions, 
a7 _ 9, BH _ 
SV. eW_. eh SW EW ow 
Su” Sedu 
se ee 
x Pa eae! Sues Sane 
This last supposition removes the indeterminateness of JV” itself, and therefore of its 
partial differential coefficients; of which, for the two first orders, eight vanish by 
VOL. XVII. H 
