730 
MR. J. LAEMOR ON A DYNAMICAL THEORY OF 
for integration through any fixed period of time. Thus'' 
d8^ dt] d^T] d^ j 
IT TtTii Tif Tit) 
■ !d^ _ ^ /^| _ d^ 
[ (Y/ \ dy dz j dy \ dz dx 
^ _ 
dh V dx 
f 1 ^ <^r 
= 0. 
On integration by parts in order to replace the differential coefficients of 8 (^, -q, 
by these variations themselves, we obtain, leaving out terms relating to the 
beginning and end of the time. 
where (/, m, n) are the direction-cosines of the element of surface dS. As the 
displacements 8 (^, -q, Q are as yet quite arbitrary, the equations of elastic vibration 
of the medium are therefore 
d^^ — 0 
^ dt~ dy dh dz dg 
^ df~ dz df ~ Tx dh ~ ^ 
^ dt~ dx dg dy df 
From them it follows that 
+ + S = 0 
dx ^ dy^ dz 
in other words, that there is no compression of the medium involved in this motion, 
whether we assume that it has the property of incompressibility or not. 
15. In accordance with the general dynamical principle, all the conditions which it 
is essential to explicitly satisfy at an interface between two media are those which 
secure that the variation of the energy shall not involve a surface integral over this 
intei-face. To express these conditions most concisely, let us take for the moment the 
* Of. G. F. FitzGerald, “ On the Electromagnetic Theory . . . ‘ Phil. Trans.,’ 1880. In that 
memoir the rotation is represented by Ttt (/, gr, 7j), instead of simply (/, g, 7i) as above, in order to he in 
line with Maxwell’s electrodynamic equations. 
