90 ROYAL SOCIETY OF CANADA 
which the strain and the unstraining may be accomplished. In other 
words, P, Q, R, etc., representing the stress components at a point at 
which the corresponding strain components are e, f, g, etc. whether 
during the straining or the unstraining, 
ih ( Pde + Qdf + Rdg + Sda + Tdb + ic) dx dy, dz 
must be an exact differential. 
The condition that this expression shall be an exact differential is that 
Pde + Qdf + Rdy + Sda + Tab + Ude 
itself shall be an exact differential, and, therefore, that P, Q, R, S, T, 
shall be proportional to the rates of increase with respect to e, f, g, a, b, ¢, 
respectively, of a function of e, f, g, a, b, c, and of these quantities only. 
If, therefore, this condition be assumed to be fulfilled, the equation of 
work done for an isolated system becomes— 
Drames ll (V4) HE) be 
V being the potential energy per unit of volume; or if we indicate by P 
the potential energy and by K the kinetic energy of the body, 
dPtdKkK=0; 
and hence P + K = constant, 
= 
the law of the conservation of energy. 
Hence the additional hypotheses which are required for the deduc- 
tion of the conservation of energy are the Third Law of Motion and the 
hypothesis that the stress components at any point are proportional to the 
rates of increase, with respect to the corresponding strain components at 
the point respectively, of a function of these strain components only. 
It is, of course, well known that, owing to the thermal changes 
uccompanying strain and the variation of the stress-strain relations with 
temperature, 
Pde + Qdf + Rady + Sda + Tab + Ude 
is not in general for actual bodies undergoing actual strains an exact dif- 
ferential; but in abstract dynamics we ignore variation of temperature 
in the body, and consider only ideal perfectly elastic bodies strained in 
such a way that variations of temperature do not occur. 
The equations of motion given above and the law of the conservation 
of energy, however, are not sufficient for the solution of even all general 
problems connected with elastic solids and fluids. It is frequently neces- 
sary to employ the equation of continuity, and it would thus appear at 
first sight that the treatment of continuous bodies requires still another 
hypothesis. The equation of continuity seems to be regarded usually as 
