THE ELECTRIC AHD LUMIHIFEROUS MEDIUM. 
721 
Action of the system in the manner of Lagrange. If these conditions appear 
to be too numerous, the reason must be either that the forcive which compels the 
observance of some constraint has not been explicitly included in the analysis, or else 
that the number of the constraints has been over-estimated. In each problem in 
which the mathematical analysis proceeds without contradiction or ambiguity to 
a definite result, that result is to be taken as representing the course of the 
dynamical phenomena in so far as they are determined by the energy as specified; 
a further more minute specification of the energy may however lead to the inclusion 
of small residual jfiienomena which had previously not revealed themselves, 
4. The object of these remarks is to justify the division of the problem of the 
determination of the constitution of a partly concealed dynamical system, such as the 
gether, into two independent parts. The first part is the determination of some form 
of energy-function which will explain the recognized dynamical properties of the 
system, and which may be further tested by its application to the discovery of new 
properties. The second part is the building up in actuality or in imagination of some 
mechanical system which will serve as a model or illustration of a medium possessing 
such an energy-function. There have been cases in v/hich, after the first part of the 
problem has been solved, all efforts towards the realization of the other part have 
resulted in failure ; but it may be fairly claimed that this inability to directly con¬ 
struct the properties assigned to the system should not be allowed to discredit the 
part of the solution already achieved, but should rather be taken as indicating some 
unauthorized restriction of our ideas on the subject. Of course where more than one 
solution of the question is possible on the ascertained data, that one should be pre¬ 
ferred which lends itself most easily to interpretation, unless some of the others should 
prove distinctly more fertile in the prediction of new results, or in the inclusion of 
other known types of phenomena within the system. 
5. In illustration of some of these principles, and as a help towards the realization of 
the validity of some parts of the subsequent analysis, a dynamical question of suffi¬ 
cient complexity, which has recently occupied the attention of several mathematicians, 
may be briefly referred to. The problem of the deformation and vibrations of a thin 
open shell of elastic material has been reduced to mathematical analysis by Lord 
Ravleigh,* on the assumption that, as the shell can be easily bent but can be 
stretched only with great difficulty, the potential energy of stretching would not 
appear in the energy-function from which its vibrations in which bending plays a pro¬ 
minent part are to be determined,—that in fact the shell might be treated as inexten- 
sible. But a subsequent direct analysis of the problem, of a more minute character,! 
led to the result that the conditions at the boundary of the shell could not all be 
satisfied unless stretching is taken into account. The reason of the discrepancy is 
* Lord Rayleigh, “ On tie Infinitesimal Bending of Surfaces of Revolution,” ‘ Proc. Lond. Math. 
Soc.,’ 1882. 
t A. E. H. Love, “ On the . . . Vibrations of a Thin Elastic Shell,” ‘ Phil. Trans.,’ 1888. 
MDCCCXCIV.—A. 4 Z 
