188 
NATURE 
[APRIL 23, 1914 
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LETTERS, TO THE EDITOR. 
[The Editor does not hold himself responsible for 
opinions expressed by his correspondents. Neither 
can he undertake to return, or to correspond with 
the writers of, rejected manuscripts intended for 
this or any other part of Nature. _No notice is 
taken of anonymous communications. ] 
The Sand-Blast. 
Amonc the many remarkable anticipations contained 
in T. Young’s lectures on natural philosophy (1807) 
is that in which he explains the effect of what is now 
commonly known as the sand-blast. On p. 144 he 
writes :—-‘‘ There is, however, a limit beyond which the 
velocity of a body striking another cannot be increased 
without overcoming its resilience, and breaking it, 
however small the bulk of the first body may be, and 
this limit depends on the inertia of the parts of the 
second body, which must not be disregarded when 
they are impelled with a considerable velocity. For 
it is demonstrable that there is a certain velocity, de- 
pendent on the nature of a substance, with which the 
effect of any impulse or pressure is transmitted through 
it; a certain portion of time, which is shorter accord- 
ingly as the body is more elastic, being required for 
the propagation of the force through any part of it; 
and if the actual velocity of any impulse be in a greater 
proportion to this velocity than the extension or com- 
pression, of which the substance is capable, is to its 
whole length, it is obvious that a separation must be 
produced, since no parts can be extended or com- 
pressed which are not yet affected by the impulse, 
and the length of the portion affected at any instant 
is not sufficient to allow the required extension or 
compression. Thus if the velocity with which an 
impression is transmitted by a certain kind of sound be 
15,000 ft. in a second, and it be susceptible of com- 
pression to the extent of 1/200 of its length, the 
greatest velocity that it can resist will be 75 ft. in a 
second, which is equal to that of a body falling from 
a height of about go ft.” 
Doubtless this passage was unknown to O. Rey- 
nolds when, with customary penetration, in his paper 
on the sand-blast (Phil. Mag., vol. xlvi., p. 337, 1873) 
he emphasises that ‘‘the intensity of the pressure be- 
tween bodies on first impact is independent of the 
size of the bodies.” 
After his manner, Young was over-concise, and it is 
not clear precisely what circumstances he had in con- 
templation. Probably it was the longitudinal impact 
of bars, and at any rate this affords a convenient 
example. We may begin by supposing the bars to be 
of the same length, material, and section, and before 
impact to be moving with equal and opposite veloci- 
ties v. At impact the impinging faces are reduced to 
rest, and remain at rest so long as the bars are in 
contact at all. This condition of rest is propagated in 
each bar as a wave moving with a velocity a, char- 
acteristic of the material. In such a progressive wave 
there is a general relation between the particle- 
velocity (estimated relatively to the parts outside the 
wave) and the compression (e), viz., that the velocity 
is equal to ae. In the present case the relative par- 
ticle-velocity is v, so that v=ae. The limit of the 
strength of the material is reached when e has a cer- 
tain value, and from this the greatest value of v (half 
the original relative velocity) which the bars can bear 
is immediately inferred. 
But the importance of the conclusion depends upon 
an extension now to be considered. It will be seen 
that the length of the bars does not enter into the 
question. Neither does the equality of the lengths. 
However short one of them may be, we may contem- 
plate an interval after first impact so short that the 
NO! 2321, 30L: 92] 
wave will not have reached the further end, and then 
the argument remains unaffected. However short one 
of the impinging bars, the above calculated relative 
velocity is the highest which the material can bear 
without undergoing disruption. 
As more closely related to practice, the case of two 
spheres of radii 7, 1’, impinging directly with relative 
velocity v, is worthy of consideration. According to 
ordinary elastic theory the only remaining data of the 
problem are the densities p, p’, and the elasticities. 
The latter may be taken to be the Young’s moduli 
q, q', and the Poisson’s ratios, 0, 0’, of which the two 
last are purely numerical. The same may be said 
of the ratios q'/q, p’'/p, and 7’/r. So far as dimen- 
sional quantities are concerned, any maximum strain 
e may be regarded as a function of 1, v, q, and p. 
The two last can occur only in the combination q/p, 
since strain is of no dimensions. Moreover, q/p=a’, 
where a is a velocity. Regarding e as a function of 
r, v, and a, we see that v and a can occur only as 
the ratio v/a, and that y cannot appear at all. The 
maximum strain then is independent of the linear 
scale; and if the rupture depends only on the maxi- 
mum strain, it is as likely to occur with small spheres 
as with large ones. The most interesting case occurs 
when one sphere is very large relatively to the other, 
as when a grain of sand impinges upon a glass 
surface. If the velocity of impact be given, the glass 
is as likely to be broken by a small grain as by a 
much larger one. It may be remarked that this con- 
clusion would be upset if rupture depends upon the 
duration of a strain as well as upon its magnitude. 
The general argument from dynamical similarity 
that the maximum strain during impact is independent 
of linear scale, is, of course, not limited to the case 
of spheres, which has been chosen merely for con- 
venience of statement. RAYLEIGH. 
The Earth’s Gontraction. 
THE conclusion of the Rev. Osmond Fisher 
(NaturE, February 26) that if the moon originated as 
a detached portion of the earth, the earth’s radius at 
the time (even allowing for the much more rapid rate 
of rotation indicated by Sir G. H. Darwin’s re- 
searches) must have been about three times its present 
one, leads to a very interesting speculation, namely, 
as to whether the earth’s radius may not have con- 
tracted very considerably within the time represented 
by the known geological formations. There is, I 
think, observational evidence which warrants us in 
believing this to have been the case. 
Prof. Heim estimated the linear compression re- 
quired to produce the Alps at seventy-four miles, 
which means a reduction of the earth’s radius by 
twelve miles, or 0-3 per cent. Taking the whole of 
the existing mountain ranges, we may roughly esti- 
mate a total reduction of ten times this amount, or 
3 per cent., as being indicated since the middle of the 
Tertiary epoch. Yielding of the earth’s crust by 
intense folding has probably always taken place in 
particular areas, but it is a fact that as a whole the 
rocks show more and more folding, faulting, and 
overthrusting the farther back we go into the geo- 
logical record, and the mountains formed in the older 
epochs have long since been removed by denudation, 
which was naturally most active where the plication 
was most intense. Taking the rock-structures alone 
into consideration, would any geologist who has 
worked extensively amongst the oldest fossiliferous 
rocks affirm that the evidences are against a contrac- 
tion of the earth’s radius of the order of 20 per cent. 
since they were deposited? A contraction of this 
magnitude would be accommodated by a continuous 
folding of the crust into anticlines and synclines at 
) angles of 37° with the horizontal, and most accounts 
