88 ROYAL SOCIETY OF CANADA 
throughout the whole volume. We thus obtain, after some purely kine- 
matical transformations : 
oP dU oT aU dQ eek oT 
Nea du + Se + 5: du + er F Sy dv + = wy = Ty dw 
os 
+ Sy dw af? 

Se Le ) dx dy dz 
+ ff e (au + Yay + Ziz) Bish 
SLY ee aay) ata ly de = 
Ds AE ASUS US TUE se ae) dt dx dy dz = 0. 
This equation may now be shown, by the aid of mathematical theorems 
and without the employment of any dynamical assumptions in addition to 
those already employed, to be equivalent to the following: 
SS (ru + Gde + Hi ) ds 
Sit / fe de + Qdf + Rdg + Sda + Tdb + rie) dx dy dz 
ai ie p (vax yg a2 lz) dx dy dz 
OPEN 
where e, f, g, a, b, c are, according to the ordinary notation, the com- 
ponents of the strain, and F’, G, H are the surface tractions at the element 
of surface dS. 
It follows, from the ordinary definition of work done,’ that the first 
term in this equation represents the work done by the surface tractions, 
ol 
! The ordinary definition of work done by a force is the product of the magnitude 
of the force into the component displacement of its point or place of application in its 
direction ; work done by a force when negative being called also work done by the 
body, on which it acts, against the force. Asa displacement is necessarily relative, 
the value of the work done, according to this definition, is arbitrary, without the 
specification of a reference system. Prof. Newcomb, for this reason (Phil. Mag. [5], 
vol. xxvii., 1889, p. 115), proposed to define work done as ‘“‘the product of the inten- 
sity of the force into the amount by which the two material points between which it 
acts approach to or recede from each other.” The following objections may be made 
to this definition: (1) It isa definition of the work done by a stress, not of the work 
done by a force; and the latter, from which the former may be deduced, is too useful 
a conception to be abolished. (2) It is not a definition merely, but involves a dynam- 
ical hypothesis as well, viz., the Third Law of Motion; and hypotheses should be 
kept quite distinct from mere definitions. (3) It is applicable only in cases of action 
at a distance, being a particular form which the ordinary definition takes in such 
cases. Prof. Lodge, on the other hand (Phil. Mag. [5], vol. viii., 1879, p. 278), has pro- 
posed the following definition: ‘‘ Whenever a body exerting a force moves in the 
