278 REPORT—1862. 
true of any matérial system. In the latter case they are not sufficient for 
the equilibrium, but all that we are concerned with in the demonstration of 
equations (26) is that they should be true. 
On the other hand, the form of V or ¢ to which MacCullagh was led is that 
of a homogeneous fanction, of the second order, of the three quantities 
dw dv du dw dv du 
dy dz dz dx dx dy’ *) 
which, as is well known, are linear functions of the similarly expressed 
quantities referring to any other system of rectangular axes. On substituting 
in (23), we see that the normal tensions on planes parallel to the coordinate 
planes, and therefore on any plane since the axes are arbitrary, vanish, while 
the tangential tensions satisfy the three relations 
—— ae T,.=—T,, Loy eg ie Se (29) 
so that the equations of moments of an element are violated. The relative 
motion in the neighbourhood of a given point may be resolved, as is known, 
into three extensions (positive or negative) in three rectangular directions and 
three rotations. The directions of the axes of extension, and the magnitudes 
of the extensions, are determined by the six quantities (24) and (25), while 
the rotations or angular displacements are expressed by the halves of the three 
quantities (28). In this theory, then, the work stored up in an element of the 
medium would depend, not upon the change of form of the element, but upon 
its angular displacement in space. 
It may be shown without difficulty that, according to the form of ¢ assumed 
by MacCullagh, the equations of moments are violated for a finite portion of the 
mass, and not merely for an element. Supposing for simplicity that the 
medium in its undisturbed state is free from pressure or tension, let us leave 
the form of ¢ open for the present, except that it is supposed to be a function 
of the differential coefficients of the first order of u,v, w with respect to 
#, y, z, and let us form the equation of moments round one of the axes, as 
that of w, for the portion of the medium comprised within the closed surface 8. 
This equation is 
\\V{-3 we van? f dry de (Ry Q:)dS=0, 
the double integrals bevy to the surface. Since all the terms in this 
equation are small, we may take w, y, 2 for the actual or the equilibrium 
coordinates indifferently. Substituting from equations (20), and integrating 
by parts, we en 
Vr@ as Bey} ayae(U{ (2) (ten) aa 
WG iv i Je (zy } ae dy he Qz) dS 
WG I ay wat (Zz) | ate dy ae=0. 
The double ee in a equation destroy each other by virtue of (22), so 
that there remains 
{7G {7 iy) Ap (z) } deay demo. s+ + + (80) 
But this equation cannot be satisfied, since the surface § within which the 
