1898 - 99 .] Prof. Knott on Earthquake Vibrations. 
577 
formula given above.* It is evident that the paths traversed by 
any vibration are at right angles to the wave-fronts and can be 
straight lines only when these are spherical, that is, when the speed 
is constant at all depths. If the speed increases with nearness to 
the centre, the paths will be convex towards the centre. In this 
case the true average speed will be somewhat greater than the value 
obtained by dividing length of chord by the time taken. Until 
many more observations have been accumulated, it would probably 
be a waste of labour to attempt any better approximation to the 
law of propagation of seismic disturbance through the earth. The 
mathematical difficulties are considerable, and we can hardly hope 
for other than approximate solution of the problem. 
The speed of propagation of an elastic wave depends on a 
particular coefficient of elasticity and on the density of the material. 
Assuming — and this seems the most plausible assumption — that 
both types of waves travel by brachistochronic paths through the 
earth, we conclude that, since the density is the same in both cases, 
the coefficients of elasticity must be influenced by the depth in 
quite different ways. The density is known to increase with the 
depth; and various formulae have been given by different in- 
vestigators. For ordinary purposes, where no depths greater than 
2000 miles are considered, we may use the formula, 
density = 2*75 + *0028 x depth in miles, 
which agrees very closely with Laplace’s historic formula. 
In other words, the density may be assumed to increase by 
per mile descent, or *28 per cent. The coefficient of elasticity which 
determines the propagation of the larger waves will therefore increase 
at nearly the same rate, whereas the coefficient of elasticity which 
determines the propagation of the preliminary tremors will increase 
at the rate of nearly 1*2 per cent, per mile descent. 
These results seem to have a distinct bearing upon the question 
of the internal condition of the earth. They indicate that the earth 
throughout the greater part of its mass is capable of transmitting 
two types of elastic waves, and is therefore an elastic solid. 
The only way to escape from this conclusion is to argue that the 
* I have since found that the problem has been mathematically worked out 
in a form convenient for application by M. P. Rudzki of Krakau in Gerland’s 
Beitrdge zur GeophysiJc., Bd. iii. 
