7 22 Proceedings of Royal Society of Edinburgh. [sess. 
Thus if, to help ideas, we look to a Boscovich curve, the distance 
between nearest neighbours for zero-equilibrium, which for brevity 
we shall call £, must be a point in which the curve cuts the line of 
abscissas with slope corresponding to repulsions for less distances 
and attractions for greater, and shows zero-force for all distances 
not less than £ J 2. 
§65. To investigate moduluses of elasticity, we must suppose the 
forces applied from without to the points on the boundary to be 
such as to produce homogeneous strain throughout the assemblage. 
The working out of this statical problem, to be given in a future 
communication, shows that the solid so constituted is not elastically 
isotropic; but that, on the contrary, it has essentially two different 
rigidities. It is in fact a cubical isotropic body with its two 
rigidities (article “Elasticity ,” . Encyclopaedia Britannica , ninth edi- 
tion, or vol. iii. of my Collected Papers) not equal. An extension 
of the investigation to include the supposition of forces not only 
between nearest neighbours, but between nearest and next nearest 
neighbours and none farther, gives of course the two rigidities 
generally not equal; but it allows them to be equalised by a certain 
definite relation between forces and variations of forces at the two 
distances £ and £ J 2. Imposing this condition, we have elastic 
isotropy; and I find the compressibility to be essentially of the 
rigidity. The solid thus constituted is therefore subject to 
Poisson’s restriction; and it will no doubt be found that this 
restriction is valid for any single equilibrated homogeneous distribu- 
tion of points, with mutual forces according to Boscovich, and 
sphere of influence not limited to nearest and next-nearest neigh- 
bours, but extending to any large, not infinite, number of times the 
distance between nearest neighbours. 
§ 66. Having thus failed to produce a solid free from Poisson’s re- 
striction, go back to the very simplest case, and try for another way 
of leaving its simplicity by which we may succeed. Try first to 
realise an incompressible elastic solid. When this is done we 
shall see, by an inevitably obvious modification, how to give any 
degree of compressibility we please without changing the rigidity, 
and so to realise an elastic solid with any given positive rigidity, 
and any given positive or negative bulk-modulus (stable without 
any surface constraint, only when the bulk-modulus is positive). 
