[MACGREGOR ] HYPOTHESES OF ABSTRACT DYNAMICS 89 
the second the work done by the internal stresses on the elements on 
which they act, and the third the work done by the body forces ; and 
it may be shown, by the aid of the Second Law of Motion, that the fourth 
term represents the increment of kinetic energy. Hence the equation 
states that the work done by external forces (surface tractions and body 
forces) is equal to the work done by the internal stresses on the elements 
on which they act together with the increment of kinetic energy. If, 
therefore, the system be isolated so as to be acted upon by no external 
forces, we have: 
SIT Pde + Qdf + Rdg E\Sda EUTUbE ve) de dy dz 
Herr | any? dy\° Gi rl À HE 
+ ff} d \ | (%) a5 ea = i df ie dy dz — 6, 
i.e., the work done by internal stresses, together with the increment of 
. kinetic energy, must be zero. 
If. after the occurrence of the small strain de, df, etc., the body be 
allowed to return to its initial state of strain, and if the values of the 
stress components during the unstraining are P’, Q', RB’, S’, 7", U’, work 
will be done by the elements against the stress components equal to— 
1 VB ( P'de + Q'df + Rdg + S'da + T'db + va) dx dy dz. 
Hence this expression will represent the work power which the body had 
gained because of the strain or the potential energy of the strain. Ina 
finite strain the potential energy will be the integral of this expression 
between the final and initial configurations as limits 
In order that the potential energy of a finite strain may be equal to 
the work done by the internal stresses during the strain, the above integ- 
ral must be equal to the integral of 
HIT (Pa. + Quf + Rdg + Sda + Tab + vie) dx dy dz 
between the same limits, whatever the series of infinitesimal strains by 

sense of the force it exerts, it is said to do work ; and whenever a body exerting a 
force moves in a sense opposite to that of the force it exerts, it is said to have work 
done upon it, or to do anti-work, the quantity of work being measured in each case 
by the product of the force into the distance moved through in its own direction.” 
This is clearly equivalent to the ordinary definition in cases of contact action, but 
not in cases of distance action. If applied in the latter cases, theorems involving 
work done and working power or energy, which have been established in terms of 
the ordinary definition, would no longer hold.. The law of the conservation of 
energy, for example, would have to be replaced by a more complex law of energy. 
Thus Lodge’s definition is as unsuitable for distance action discussions as New- 
comb's for those involving contact action. The ordinary definition has, therefore, 
the great advantage over both of being equally applicable to both classes of prob. 
lems. ‘See a note on the ‘Definition of Work Done” in “ Trans. Nova Scotian 
Institute of Science,” vol. viii., p. 460.) 

