276 REPORT—1862. 
where J, m, » are the direction-cosines of the element dS of the surface which 
bounded the portion of the medium under consideration when it was in its 
undisturbed state. This expression leads us to contemplate the action of the 
surrounding medium as a tension having a certain value referred to a unit of 
surface in the undisturbed state. If P,Q, R be the components of this tension 
parallel to the axes of w, y,z, they must be the coefficients of du, év, Sw under 
the sign \ so that 
[du (du du 
rav(t) +n) (2) 
(de (de [, 
Q=/f (52) +mp() +nf(Z) es ad rae ae (22) 
[dw (dw (dw 
naif) +" (Gy) +9 (ze 
These formule give, in terms of the function ¢, the components of the 
tension on a small plane which in its original position had any arbitrary 
direction. If we wish for the expressions for the components of the tensions 
on planes originally perpendicular to the axes of «, y, z, we have only to put 
in succession /=1, m=1, n=1, the other two cosines each time being equal 
to zero. If then P.,, ie T_, denote the components in the direction of the 
axis of w of the tension on planes originally perpendicular to the axes of x, y, z, 
with similar notation in the other cases, we shall have 
du __ pf dw cl afdy 
AG) TMG) TF(z)| 
(de _ pf{du __ pp {dw | 
v,=f (7) T,=f =) T= (ae) > 7 (23)* 
dw , (dv du 
rz) Tafa) (G). 
The formul hitherto employed are just the same whether we suppose the 
disturbance small or not; and we might express in terms of P_, Te &e. (and 
therefore in terms of ¢), and of the differential coefficients of u, v, w with 
respect to a, y, and z, the components of the tension referred to a surface 
given in the actual instead of the undisturbed state of the medium, without 
supposing the disturbance small. As, however, the investigation is meant to 
be applied only to small disturbances, it would only complicate the formule 
to no purpose to treat the disturbance as of arbitrary magnitude, and I shall 
therefore regard it henceforth as indefinitely small. 
On this supposition we may expand ¢ according to powers of the small 
quantities ~, &e., proceeding as far as the second order, the left-hand 
member of (19) being of the second order as regards u, v, w. The formule 
(22) or (23) show that ¢ will or will not contain terms of the first order 
according as the undisturbed state of the medium is one of uniform constraint, 
or of freedom from pressure. 
In Green’s first theory, and in the theory of MacCullagh, ¢ is supposed not 
to contain terms of the first order. Accordingly in considering the poimt 
with respect to which these two theories are at issue, I shall suppose the 
* These agree with Professor Haughton’s equations at p.100, but are obtained in a 
different manner, 
