80 
REPORT—1847. 
essential to ascertain to some degree of approximation, the velocities of pro* 
pagation through such solid masses as constitute, or maybe presumed to 
constitute generally the solid crust of the globe. To explain the relation 
between these velocities and the elasticities of the substances through which 
the waves are propagated, and the nature of the experiments by which the 
elasticities must be determined, it will be necessary to enter ititn some de¬ 
tails respecting the elementary notions on which the mathematical theory of 
vibratory motions is founded. The theory I here speak of is independent of 
^1 hypotheses on the nature of molecular action. It is given by M. Cauchy 
in various articles in his * Excrcices Mathfmatiquea*.’ 
I have already explained (aj-t.20) that, for every point of a continuou* 
solid mass in a state of constraint, there are three principal dirtciions and 
three principal tensions. The fundamental hypothesis on which SI. Cauchy 
lias founded his investigation of the differential equations of vibratory motion 
in solids, is that the principal tensions arc proportional, partly to the linear 
and partly to the cubical expansion. Thus, if c„ e, denote the linear ex¬ 
tensions at any point in the principal directions, v the cubical expansion, 
and p^, p-, yjj the principal tensions, we shall have 
^,=Ac, + Kv, 
where k and K are two constants to be determined by experiment. These 
two constants, or others equivalent to them, will necessarily enter into the 
differential equations of motion. If, instead of introducing k and K, we in- 
troduce A and B where “ 
A=A-1-K, 
B=4, 
Sg the'^St’y fo*™ (adopting the usual notation and call- 
, B d/dvdwW 
? dz{d.'^dp^^i +7 
riso pmved that distinctive characters are proved. It« 
- St- 
we must determine those of land Consequently, in any experiment, 
which may be instituted for the purpL of thus determining these velocities, 
o. ssSiS 
