740 
MR. A. McAULAY ON THE MATHEMATICAL 
conceptions of stress seems to me quite unnecessary on general grounds. It is well 
known that to every magnetic distribution there is an analogous conceivable distribu¬ 
tion of ordinary statical electricity. In the ordinary action-at-a-distance theories the 
mutual mechanical effects of different parts of a magnetic system would be exactly 
the same as the corresponding effects in the analogue. Why, then, should it be con¬ 
sidered unnecessary in the case of electrostatics, but necessary in the case of magnetics, 
to postulate a molecular couple ? Why not, in other words, say that the stress which 
Maxwell would suppose existent in the electric analogue is exactly the stress really 
existent in the magnetic system ? In the second place, although the process seems 
viciously needless, we may, if we like, conceive any physical phenomena involving 
stress as causing a molecular couple which is exactly balanced by a stress-couple. [It 
must be so equilibrated in order to insure against infinite angular acceleration of an 
element of matter—supposing, of course, that the ultimate constitution of matter 
were not heterogeneous.] This latter stress-couple will be entirely of the nature of a 
reaction, since (former paper, p. 108) it is entirely independent of the potential energy 
of strain. In the present case, then, in which we suppose electromagnetic phenomena 
to produce stress, we shall have one stress exactly equilibrating another stress, neither 
of them having anything to do with the Lagrangian function. This is only another 
way of saying that no physical conception whatever is gained by the supposition that 
the particular physical phenomenon produces a stress-couple. We shall, then, consider 
it necessary to compare our results only with the pure part of the stress which 
Maxwell supposes to exist. 
Thus in §641 Maxwell arrives at the conclusion that the stress required to pro¬ 
duce observed electromagnetic phenomena is v where 
8ttvoj = - 2H / SwB' + coK' 1 = 8ir{<f>}a> + V(YB'H'.w) . . . (26), 
where {<£'} denotes the pure stress given by 
8tt{ f }w = - H'SwB' - B'SwH' -f wH' 3 = - YB'odT - 47rwSl'H' . (27). 
Now, what in the former paper (p. 108) was called the couple stress part of v, namelv, 
V (VB'H'. w)/8tt, produces a couple per unit volume YB'H'/Itt = VI'H', and this 
must be equilibrated by some other couple per unit volume even when the body is 
not in equilibrium. This couple can only result from a couple stress, Yew, which 
produces a couple, 2e, per unit volume ; and this is quite independent of the potential 
energy of strain, and therefore of the Lagrangian function. Thus, 2e + A I H = 0. 
If now, in addition to the stress uw we take account of the reactionary stress Yew, 
we simply get the pure stress (<£'}w. We shall then merely compare the stress (pure) 
which flows from the present theory with {< f 7}. 
68. It is well-known that Maxwell’s stress can only be looked on as a normal 
