242 REPORT—1885. 7 
ete. If we suppose the whole motion to be harmonic and of period 7, then 7 
the equations become | 
_ my 
aa a = C, (G = a) 7 C, (x = &o) . 2 4 (83) 
etc., from which the motion of the various shells can be determined. 
The system will represent Helmholtz’s theory if we suppose the viscous 
terms in his expression to vanish, and consider only a single shell. The 
solution in the general case is carried further by putting : 
Ni; 
oC a? C;- Cia] 
. (84) 
and Ue — Cx; | 
i BB tO 
The equations may then be written— 
C”) 
Uy, — a, —_— — 
Uo 
i (85) © 
a8 C;? 
Uy = Ag — — 
us 
etc., whence we find w, as a continued fraction. 
By differentiating these expressions with reference to r~?, and writing - 
6 for 53 We find— 
2 7 
piel atl (1) mie + (CerCiee) Misa +, etc...  . (86)) 
Wir] U4 1 Ui-2 
Hence 
du; i ? 
= Se sama? $M, 274, e . . i : .. (8738 
Thus wu decreases as 7 increases, and if we start from 7, a small quan- — 
tity, the w’s are all large and positive ; hence alternate shells are moving — 
in opposite directions, and the motion of consecutive shells rapidly 
decreases. | 
As 7 increases the w’s decrease, and after a time one will become 
negative, passing through zero—it can be shown that w, is the first one 
thus to become negative. This gives the first critical case in the solution, — 
for then «, is infinitely great compared with £, and the solution fails. 
This equation can be put into the following more convenient form— 
etry | r? k,7R, ko?Ro \ > 
Sire [cet aD 
1 
or 
pe 
where kj, ka, etc., are the critical values of 7, and R,, R,, etc., represent 
the ratio of the energy of the several shells to the whole energy of the 
system. : 
To apply this to the motion of the ether in a transparent body, let 
m,/47*, etc., represent the whole mass contained within shell No. 1 per 
unit vol., let p/4x? be the density, and e/47? the rigidity of the ether, and 
suppose the first shell, of mass m, to be connected by a spring to a massless 
