222 REPORT—1885. 
Helmholtz supposes, ‘um die Bewegungsgleichungen zu vervoll- — 
stindigen,’ that A is proportional to the relative displacement of the 
ether and atoms in the element of volume, and writes, therefore, 
A=/p2(U — u). 
Lommel supposes that the action ‘follows Newton’s law of friction,’ 
‘and depends on the relative velocity of the two; he puts, therefore, 
A= ed (U—u). 
dt 
The expression given by Voigt is much more complicated, and can 
best be considered later. Thus the equations we have to deal with are— 
2 2 
m du _ 2 + 6?(U —w) 
C dz? 
oi 20 
LU 22 2 dU ( ) 
bog =P (U—u)-a te 
d ¢ 
(Helmholtz), and 
Ou tg thn poder 
sist a a age rashes u) 
(21) 
?U ne rey? d = _ 2 9 dU * 
La pa. U uv) Aged as 
(Lommel). 
The method of solution is the same in both. w and U, which, strictly 
are the displacements of ether and matter in the same volume in the dis- 
placed condition, are treated as if they were the displacements of ether 
and matter having the same undisturbed co-ordinates x, y, z. This is 
legitimate, for U and wu are both taken to be functions of the position of 
the wave front and the time only, and hence for all points on the same 
wave front U has at a given instant the same value. 
Assume, then, 
u“u= We —kz+ in (z—ct)/e 
U = Ae7k=t+in (—ed/c+ ) (22) 
i is the coefficient of absorption, c the velocity, and 27/n the period of 
the vibration. . 5 
On substituting these valnes in Helmholtz’s equations, we find 
1 Bim B pe uae Bley 1) 
5 fx B p pn a p F (say) (23) 
bs a 
ce on? at «an? = an? (un? — a? — B?)? + yn 
and 
2k pty? 1 
cn an (pn? — a? — (37)? + An? 
== G,(87;) |! soem aan 
* In this equation the sign of 8? has been changed from that given b i 
é y Lommel in 
accordance with the remark on p. 221; but see Lommel’s repl 6 iot. Wi j 
fabled ply to Voigt, Wied. Ann. 
t A, of course, no longer has the same meaning as above, but is the amplitude of 
the matter vibrations. 
