722 
MR. J. LARMOR OR A DYNAMICAL THEORY OF 
that, if the question is simplified by taking’ the shell to be inextensible, a static 
extensional stress ought at the same time to be recognized as distributed all along the 
surface of the shell, and as assisting in the satisfaction of the necessaiy conditions at 
its free edge; the stress-condition that can be adjusted in this manner may thus be 
left out of consideration, as taking care of itself. If we suppose the shell to be not 
absolutely inextensible, this tension will be propagated over the shell by extensional 
waves with finite but very great velocity; it will therefore still be almost instan¬ 
taneously adjusted at each moment over a shell of moderate extent of surface, and the 
extensional waves will thus be extremely minute; such waves would have a very 
high period of their own, but in ordinary circumstances of vibration they would be 
practically unexcited. These remarks appear to be in keeping with the explanation 
of this matter which is now generally accepted. 
6. The dynamical method as hitherto explained applies only to cases in which the 
forces are all derived from a potential-energy function, or are considered as explicitlv 
applied from outside the system; in the latter case they may be, as yon Helmholtz 
remarks, any arbitrary functions of the time. By ineans of the Dissipation Function 
introduced by Lord Rayleigh, the equation of Varying Action will be so modified as 
to include probably all the types of frictional internal forces that are of much 
importance in physical applications. 
7. A few words may be said with respect to notation. In order to reduce as much 
as possible the length to which formulse involving vector quantities extend themselves 
in ordinary Cartesian analysis, a vector will usually be specified by its three Cartesian 
components enclosed in brackets, in front of which may be placed such operators as 
act on the vector. Of particularly frequent occurrence is the operator which deduces 
the doubled rotation of an element of volume from the vector which represents the 
translation ; this will, after Maxwell, receive a special designation, and will here be 
called the vorticity or curl of that vector. If the vector represent the displacement 
in an incompressible medium, i.e., if it has no convergence, we have (curl)“ = — V’, 
where is Laplace’s well-known scalar operator. The introduction of still more 
vector analysis would further shorten the formulae, and probably in practised minds 
lead to clearer views; but the saving-would not be very great, while as yet facility 
in vector methods is not a common accomplishment. In the various transformations 
by means of integration by parts that occur, after the manner of Green’s analytical 
theorem, it is not considered necessary to express at length the course of the analysis; 
so as there is no further object in indicating explicitly by a triple sign the successive 
steps by which a volume integration is usually effected, it will be sufficient to take 
the symbol c/r to represent an element of volume and cover it by a single sign of 
integration. In the notation of surface integrals, the ordinary usage is somewhat of 
this kind.’’" 
* Various mattei’s have been treated from rather different points of view in the abstract of this 
paper, ‘Roj. Soc. Proc.,’ voh 54, pp. 438-461. 
