[ MACGREGOR ] HYPOTHESES OF ABSTRACT DYNAMICS 87 
to be discarded is the one found to be inconsistent with fact, should it be 
possible to carry out a satisfactory crucial experiment, or, failing such 
experiment, the one found least useful in co-ordinating the phenomena of 
elasticity. 
Whether or not I am right in thinking Mr. Love’s position on this 
point untenable (and [I shall return to this question later on), there can 
be no doubt that, since the conception of contact action is now widely 
applied, a statement of the hypotheses of dynamics ought to be such 
as would be applicable to cases in which bodies are regarded as con- 
tinuous and their elements as acting directly only on elements in contact 
with them. It is, therefore, desirable to consider what modifications 
must be made in the above statement of hypotheses, in order to adapt 
them to the discussion of problems involving contact action. 
The hypotheses ordinarily employed in the discussion of elastic solids 
on the contact-action hypothesis are Newton’s Second and Third Laws 
and the law of the conservation of energy, As I have pointed out in the 
paper referred to above, however, the conservation of energy involves the 
Second Law of Motion, and is not, therefore, independent of it, the Second 
Law being necessary to prove that 3 mv? is equal to the kinetic energy. 
However convenient, therefore, for practical application, these laws form 
an illogical, because redundant, statement of the hypotheses employed. 
We have, therefore, to ask what independent hypotheses, in addition to 
Newton’s Second Law, will enable us to obtain equations determining the 
motion of an elastic solid, including the law of the conservation of energy. 
What these hypotheses are will be apparent from the following sketch 
of the reasoning by which this law may be obtained. It will be found in 
any treatise on elasticity that from Newton’s Second and Third Laws 
alone we obtain the following as equations of motion : 

{ Olle oU ro) Sf sy | 
Phy 20) ns 

ù >0\ | 
reg nes ty (v= pda dy dz = 0, 
~ 
a 

OM OSs Sle d w\ |. 
ls ay tbe + p (2-5) ANR 1e 
where P, Q, R,S, T, U, are, according to the ordinary notation, the stress 
components at the point whose unstrained co-ordinates are x, y, 2, and 
whose displacements are u, v, w; X, Y, Z, are the components of the body 
forces (per unit of mass) exerted at this point, and pis the density. To 
find the work done during the small increment of the strain, by which 
the component displacements u, v, w, are increased by du, dv, dw, we mul- 
tiply these equations by du, dv, dw, respectively, add them, and integrate 
