92 ROYAL SOCIETY OF CANADA 
The other equations required for the solution of problems in elastic 
solids and fluids, such as equations giving the relation of the stress com- 
ponents to the strain components, the relation of pressure to density, 
etc. are furnished by physical dynamics. So far as abstract dynamics is 
concerned, they are merely definitions of the kind of body with which 
we may be dealing. 
We have thus seen that, in the discussion of perfectly elastic bodies 
of invariable temperature, by means of the conception of contact action, 
in order to obtain the equations of motion, the equation of continuity and 
the conservation of energy, we employ two hypotheses, in addition to 
the Second Law of Motion, viz., the Third Law of Motion and the hypo- 
thesis 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. 
These two hypotheses together, it will be noticed, constitute a general 
specitication of stress, the Third Law asserting the equality and opposi- 
tion of the forces which constitute it, the other hypothesis stating upon 
what the magnitude of these forces depends. They may thus be con- 
veniently combined in one by defining a stress as a pair of equal and 
opposite forces acting between two contiguous elements of a body and 
estimated per unit of area of the surface across which they act, and by 
asserting that the stresses developed in an elastic body by a strain depend 
upon the strain in the way just specified. 
Hence, in cases in which the conception of contact action is employed 
in abstract dynamics, as well as, according to the conclusion reached in 
the paper cited above, in cases in which forces are regarded as actions at 
a distance, the hypotheses may be considered to be two, which in cases of 
contact action are as follows : 
(1) Lhe Law of Force: Newton’s Second Law of Motion. 
(2) The Law of Stress: The stresses between the elements of a con- 
tinuous body, which are called into play when it undergoes strain, are 
such that their components are proportional to the rates of increase, with 
respect to the corresponding strain components respectively, of a function 
of these strain components only. 
In the treatment of elastic solids and fluids the internal stresses are 
frequently regarded as contact actions, while the body forces are regarded 
as distance actions. In such cases, if Newton’s Second Law be employed 
as one hypothesis, the law of stress will require to be applied in its two 
forms—that suitable for contact actions, and that suitable for distance 
actions. But in any case which is treated as a case of purely contact 
action, or of purely distance action, only the appropriate law of stress 
will be required in addition to Newton’s Second Law. 3 
It will be noted that the law of stress in the case of contact action is 
more complex in expression than the corresponding law in the case of 
