612 
Proceedings of the Royal Society 
balanced system of forces applied to a connected system of points, 
is capable of resolution into three rectangular systems of parallel 
self-balanced forces applied to the same points.” 
Let X, &c., be the forces resolved parallel to any three ortho- 
gonal axes ; find the six sums or integrals, ^X#, y, %Zz, 3Yz = 
%Z y, %Zx = 2,Xz, Xy = ; these are called the “ rhopimetric 
coefficients.” Conceive the ellipsoid of whose equation these are 
the coefficients ; then for the three axes of that ellipsoid (called 
the “ isorrhopic axes”) each of the last three coefficients is null ; 
and the three systems of forces parallel respectively to those three 
axes are separately self-balanced. 
The theorem may be extended to systems of moving masses by 
d 2 x 
putting X-m- &c., instead of X, &c. If for any system of 
forces, the last three rhopimetric coefficients are null, and the first 
three equal to each other, every direction has the properties of an 
isorrhopic axis. This, of course, includes the case in which all 
the coefficients are null ; and in that case the system of forces is 
said to be “ Arrhopic.” The author shows that the six rhopimetric 
coefficients of a system of forces externally applied to an elastic 
solid, being divided by the volume of the solid, give the mean 
values throughout the solid of the six elementary stresses. Those 
are called the “ Homalotatic stresses.” 
If we calculate from them the corresponding externally applied 
pressures, these may be called the 1 ‘ Homalotatic pressures.” 
Take away the homalotatic pressures from the actual system of 
externally applied pressures, and the residual pressures will be 
arrhopic ; that is to say, their components parallel to any direction 
whatsoever will be separately self-balanced, and may have their 
straining effects on the solid separately determined ; and hence, 
the axes to which those residual pressures are reduced may be 
arbitrarily chosen, with a view to convenience in the solution of 
problems. 
The author then demonstrates that those problems respecting 
the distribution of stress in an elastic solid, in which the stresses 
are expressed by constants and by linear functions of the co-ordi- 
nates, are all capable of solution independently of the coefficients 
of elasticity of the substance. 
