ON OPTICAL THEORIES. 229 
Lommel, let us assume, according to this third supposition of Ketteler’s, 
that the reaction between the ether and matter is proportional to the 
relative accelerations of the two. Helmholtz supposes it proportional to 
the relative displacements, Lommel to the relative velocities. In this 
case, then, 
a9 a? 
A=- 2 w-, 
and hence 
a) 
m qt P Gp is, 
s PU 2 a? pr i 
dt? / dt? ( i Siernlat 
Thus 
us « Bm dU_ mx 
di? m+p d® m+ p? ; ; + Gm 
ape =d2u ‘ i mm 
pp Pt d@—p—p? . . . (46) 
And, with Ketteler’s assumptions as to the forces X and 2, these may be 
written as follows— 
du (PU wy gu 
me + pe ioba 12 
Pu, &U a aepe ag m= Se 
vad @ Sieedae qt aS 
uO oa + Bap (#U +87), 
which are the same in form as Ketteler’s equations, though a? is not the 
rigidity of the free ether, while there is a relation between C and C’, for 
Gy at a Le 
po m+ (3 
48 
p— PP? 
However, this does not matter, for it is the product CC’ which comes 
into the fundamental equations of the solution, and we find 
2 
D (Aeoach)or «:) 
_ ioe (m3 
(x3 ) di?) 
nn 
2k m ais 50 
cna? 2 1 2 as ‘ ( aay) 
(aw “ck. Ping 
where D = CO’, and K is proportional to y?. 
The quantity a?/m is no longer the square of the velocity in free 
space, and cannot be put equal to unity, and, in fact, a?/m will be the 
square of the refractive index for very long waves. Ketteler (p. 398) 
