2 
| MACGREGOR ] HYPOTHESES OF ABSTRACT DYNAMICS 93 
distance action. This arises from the fact that, as pointed out in my 
former paper, the law of stress, which is very generally, if not univer- 
sally, employed in cases of distance action, is more than sufficient for the 
deduction of the conservation of energy. In such cases, if S,,. S,., S,,, ete. 
be the stresses between the particles, m, and m,, m, and m,, m, and m,, etc., 
and if S,,, 8x 8» ete., be the distances between these pairs of particles 
respectively, the deduction of the conservation of energy requires only 
that 
S,,ds,, stg $,;45,5 2 S,,d8,5 + etc., 
shall be a complete differential, which again requires only that the stresses 
shall be proportional to the rates of change, with respect to the corre- 
sponding distances, of a function of the distances of all the pairs of attract- 
ing or repelling particles of the system. But it is generally assumed that 
each stress is a function of the distance between the particles between 
which its acts, and of this distance only, This assumption makes each 
term of the above expression an exact differential. It was shown in my 
former paper that the additional assumption thus introduced, over and 
above that required for the deduction of the conservation of energy, may 
be regarded as the “physical independence” of stresses, 1. e., that the 
stress between any two particles of a system is independent of the stresses 
between them and other particles. 
Were it not for the employment of this additional assumption in pro- 
blems treated as cases of distance action, it would be possible, by à suit- 
able extension of the definitions of stress and strain, to express the law of 
stress in a way which would be applicable both to the contact action 
and to the distance action modes of treatment. Such simplification, how- 
ever, would be of doubtful utility. But it is obviously of some import- 
ance that we should recognize that if Newton’s Second Law be employed 
as one of the hypotheses of dynamics, then, whether we employ the 
conception of action at a distance or of contact action, only one other 
hypothesis is necessary, and one other is sufficient, for the purposes of 
abstract dynamics, viz., one which gives a general specification of the 
stresses which occurin nature, And it is of importance also to notice that 
in applications of the molecular conception of the constitution of bodies, 
an additional assumption has generally been employed over and above 
what. are necessary for the deduction of the general equations of motion 
and of the law of the conservation of energy. 
To sum up, then, the following may be said to be a complete state- 
ment of the independent hypotheses necessary and sufficient to give the 
general equations of motion and the law of the conservation of energy, 
Newton’s Second Law of Motion being taken to be one of them : 
I. In CASES oF Contract AcTION. 
(1) The Law of the Constitution of Bodies.—Bodies (including media 
by which bodies commonly so called may be supposed to be connected) 
