336 Proceedings of the Royal Society 
due to parts in the primary potential, which are respectively of the 
degrees 2, 3, 4 . . . in the strains. The first of these has been 
completely investigated in the present paper, and the potential is 
shown to depend on two invariants which are functions of the 
secondary strains, and of six quantities called the primary quasi- 
strains. In fact, borrowing a term from the theory of reciprocal 
surfaces, we may say shortly that the part of the potential energy 
under consideration is + n J 2 ^ , where is the invariant 
of the first order of the secondary strains and primary quasi-strains, 
and - J 2 is the corresponding invariant for the reciprocals of these 
systems. 
It is also shown in the present paper that these quasi-strains 
play an important part in the elasticity of isotropic solids ; for 
besides the above result, it appears that, with the limitation of the 
potential already mentioned, the products of the stresses into the 
strained element-volume are directly expressible in terms of them, 
and that the principal axes of stress coincide with those of quasi- 
strains. The present paper also contains the equations which 
express the small motions of a strained solid, with the view of 
testing whether they present any analogy with the luminous 
waves. The results are negative, as was to be expected, there 
being in glass three real waves for every direction of the wave- 
front, and the wave surface being of the sixth class. In the case 
where the primary stress is symmetrical round an axis, an ellipsoid 
of revolution detaches itself from the general surface. 
Among other subsidiary results of this paper may be mentioned 
the derivation (from the law of superposition of strains) of the 
symbolical expressions for the stresses in terms of the strain- 
variations of the potential energy (already found in another shape 
by M. Boussinesq), and the symbolized solution of the converse 
problem. 
The law of resolution of strains and quasi-strains has been shown 
to be identical with that of stresses and with various other mathe- 
matical magnitudes, among which may be mentioned the system 
consisting of the moments and negative products of inertia of a 
solid body. A general view of this law of resolution of stresses is 
given and coupled with a parallel view of forces, along with a 
