ON THE THEORIES OF ELEVATION AND EARTHQUAKES. 75 
gated. I shall therefore commence the following explanations with this 
hypothesis. It should also be understood that the results deduced from the 
mathematical investigation of the problem depend on the assumption that 
the displacement of each vibrating particle from its place of rest, is very 
smaii) an assumption which may be considered as true to u sutheient degree 
of approximation in such vibiiitory motions as those frequently experienced 
in modern eartliquakcs, except at points so near the focus from u hicli they 
proceed as to render the exception of little importance. 
35. PrnpaguHon of X^brationsaloug a Cifliitdrical Tuftc.—ThtM’ibrations 
transmitted through fluids are more simple than those tranamitted through 
solids. I shall therefore begin with the former case, and the explanations 
will perhaps be still further simplitieil if we oouueiru the fluid to be an 
plastic one, as atmospheric air. I shall also first take the case in which the 
vibrations are propagated along a cylindrical tube. Let us them suppose 
the tube AB of indefinite Icngtli to be filled with atmospheric air, and 
conceive a small disturbance of any kind instantaneously communicated in 
any manner to the portion of the fluid occupying the space yjy bounded by 
ti^sveKe sections perpendicular to the axis of the tube, and then suppose 
the fluid to be left entirely to itself. In a very short time the vibratoiy 
Fig. 10. 
t - 1 _ »_ f _ y r 
A___B 
motion will be, entirely transferred from tlic particles originally iHsturbed, to 
others on the right and left, Iho particles first disturbed being left com¬ 
pletely at rest. Let p'r/ and be the portions in a state of vibration at 
any timet. Each of these spaces may be termed a rtvz<ic*, and each will 
have the satne properties, the one being propagated to the right and the 
other to the left. Generally they will ho of the same length. The particles 
beyond q' and respectively will not have begun their vibrations, and all 
those between p' and q, will liavo performed their vibratioiiH and returned 
to 3 state of rest. 
0‘) length p’g' (as:/) of the wave will be constant. 
(2.) The velocity (V) with vvliich the wave will pass from one point to 
another (the veheity of propagation) is constant, and depends on the elasti¬ 
city of the air. 
(3.) Each particle will vibrate in succession exactly in the same manner, 
fho time during which it will continue in motion, or that required for the 
wave to pass over it, and is the same for each particle in succession. 
0-) The extent through which each particle move.** in its vibration (the 
^pitfude of vibrati07i) U by hypothesis extremely small compared with tlie 
of the wave 5 it will depend on the original disturbance. The di- 
of vibration will, at a sufficient distance from the original place of 
uwturbance, be parallel lo the axis of the tube, or jierpendicular to the ante- 
j^r and posterior boumling surfaces of tlie waves, those bounding surfaces 
y? sections of the tube perpendicular to its axis, 
waves of a more eomplicated character have properties, as we shall see, 
• This tenu hu nsnaJly a more restrirted atirt dctenjiinatc raesning with reference to 
mcitory motions of this kind, hut in o\u immediate apwlicatitm of the theory of vibrations 
WV .h u wh he gcnerallv med in tho sense dclinc<J in the text. 
c snail have Uitle concern with tliat aucccssion irf vibrstiona of a peculiar type with which 
we are principally occupied ia acoustics. 
