742 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
and the second one yields a bodily forcive 
(a'2V‘-V3^, 
Both of these fot cives satisfy the condition of being null when the medium is devoid 
of rotation. But, as in the motion of a train of plane waves of length X the operator 
is replaceable by the constant — (27r/X)”, we see that the first forcive merges in 
the ordinary rotational forces of the medium, only altering its efiective crystalline 
constants in a manner dependent on the wave-length ; while the second forcive alters 
the character of the equations by adding to the right-hand sides terms proportional 
to Tj, and so modifies the wave-surface. If with MacCullagh we had taken the 
last and most general type of terms, which are not symmetrical with respect to the 
principal axes of optical elasticity, the observed dispersion of tbe optic ajces of crystals 
would clearly have been involved in the equations. The nature of the proof of 
MacCullagh’s general proposition is easily made out from the examination here 
given of this particular case. 
32. The question has still to be settled, whether the postulate of complete fluidity 
as regards irrotational motion limits the form of Wo to the one assumed by 
MacCullagh. It will I think be found that it does. For the final form of the 
varnation of the potential energy is 
S |W ch = J{. . .} dS + J (P S/+ Q -f R ^h) dr, 
where (P, Q, Pt) involve {/, g, h) linearly, but with differential operators of any orders. 
We may change it to 
8 JW dr = J[. . c/S - I curl (P, Q, R) S (^, rj, Q dr, 
the expression in the integral representing a scala.r product; and this form shows 
that the bodily forcive in the medium is curl (P, Q, R). It also shows that the curl 
operator persists on integration by parts. Now this forcive is linear in (^, 77, {), and 
taking for a moment the case of an isotropic medium, it must be built up of invariant 
differential operators. The complete list of such operators consists of curl, conver¬ 
gence, and shear operators, and their powers and products ; and these operators are 
mathematically convertible with each other. Any combination of them, operating on 
(^, 7), (,), which involves curl as a factor, will limit the medium, as has been already 
seen, to the propagation of waves only rotational; but in order to secure perfect 
fluidity as regards irrotational motions it is necessary also that the surface tractions, 
involved in the surface-integral part of the variation of the energy, shall not depend 
on the shear or convergence of the medium. Now in arriving at the final form of the 
variational equations, by successive integrations by parts, if a convergence or shear 
occur in either factor of a term in W, it will emerge at some stage as an actual conver- 
gence or shear of the medium in a surface-integral term, indicating a surface traction 
