742 
MR. A. McAULAY ON THE MATHEMATICAL 
For an isotropic body, of which that part of the Lagrangian" 
function per unit mass, which causes the body to differ from 
vacuum, is unaffected by strain [ . (29). 
2f = (//.X'Al-- 4-K.A 1 d' 2 ) - SS (t ',Vl + rr\VV) 
+ 2V{/( ),VT-«r'( )„VTjJ 
It is scarcely necessary to say, that of course it is not meant to be here implied 
that there is any body whatsoever whose general Lagrangian function—whether 
per unit mass or per unit volume—is even approximately unaffected by strain. It is 
only for brevity that we verbally contemplate such a body. There seems little or no 
reason for choosing one rather than the other of these two stresses as the normal 
type of stress for fluids. , Both of them would satisfy the condition that for a gas 
which, however large its strain, always behaved like a vacuum, the normal stress 
would be the vacuum stress which resulted from identical values of H and d'. As 
the stress of equation (29), however, agrees more closely with the pure part of 
Maxwell’s stress than that of equation (28), we will call the stress of (29) the 
normal stress for fluids. 
For solids it is harder to find a suitable normal stress, but as by far the greater 
number of them (non-magnetic bodies) behave magnetically approximately like a 
vacuum, it seems to me that the most suitable is obtained by supposing CO 0 of 
equation (27), § 55 to be zero. In this case, of course, we assume that for solids the 
normal type of stress is the stress that, with identical values of H' and d' would exist 
in a vacuum. Thus 
For a vacuum, fo) = — 27tK 0 -1 dead 7 — yfl' a>H'/8 tt (30), 
but it is needless to say that this is not perfectly satisfactory. The question may be 
asked why, in the present case, the stress of equation (29) should not be still retained 
as the normal one? The answer is, that the equation [(30), § 55] from which it is 
derived, and which actually must, in every exact discussion, be taken in its place, is a 
wholly unsuitable one for a solid, while equation (27), § 55, is a suitable one. 
Again it may be asked, Why not retain the stress equation in its original form 
(f>' = — 2m” 1 for solids ? The answer to this is, that the important fact that 
the great majority of solids behave magnetically like a vacuum is not thereby readily 
taken account of. 
69. To compare these stresses and their effects with Maxwell’s, it must first 
be noted that Maxwell has only investigated the electrostatic part of his stress 
for the case of a series of charged conductors surrounded by a dielectric that behaves 
electrostatically like a vacuum. I consider myself at liberty then to substitute any¬ 
thing for the electrostatic part of his stress which reduces to his for that particular 
case. The stress he obtains in Chapter V. of Part I. of his treatise is — 27 tK 0 -1 d' ( ) d'. 
For the particular case mentioned this may he written 
