260 REPORT—1862. 
one arbitrary constant in the expressions for the velocities of propagation 
after satisfying the numerical values of the three principal indices of refrac- 
tion, by a proper disposal of which the objections which have been mentioned 
may to a certain extent be lessened, but by no means wholly overcome. 
I come now to Green’s theory, contained in a very remarkable memoir “ On 
the Propagation of Light in Crystallized Media,” read before the Cambridge 
Philosophical Society, May 20, 1839*, and accordingly, by a curious coinci- 
dence, the very day that Cauchy’s second theory was presented to the French 
Academy. Besides the great interest of the memoir in relation to the theory 
of light, Green has in it, as I conceive, given for the first time the true 
equations of equilibrium and motion of a homogeneous elastic solid slightly 
disturbed from its position of equilibrium, which is one of constraint under a 
uniform pressure different in different directions. In a former memoiry he 
had given the equations for the case in which the undisturbed state is one 
free from pressuret. When I speak of the true equations, I mean the equations 
which belong to the problem when not restricted in generality by arbitrarily 
assumed hypotheses, and yet not containing constants which are incompatible 
with any well-ascertained physical principle. It is right to mention, however, 
that on this point mathematicians are not agreed; M. de Saint-Venant, for 
instance, maintains the justice of the more restricted equations given by 
Cauchy §, though even he would not conceive the latter equations applicable 
to such solids as caoutchoue or jelly. 
In these papers Green introduced into the treatment of the subject, with 
the greatest advantage, the method of Lagrange, in which the partial differ- 
ential equations of motion are obtained from the variation of a single force- 
function, on the discovery of the proper form of which everything turns, 
Green’s principle is thus enunciated by him :— In whatever manner the 
elements of any material system may act on each other, if all the internal 
forces be multiplied by the elements of their respective directions, the total 
sum for any assigned portion of the mass will always be the exact differential 
of some function.” In accordance with this principle, the general equation 
may be put under the form 
; P ad a? , 
\\\ pdx dy dz Ge but oe subse é v)=({ feo dy dz bo . (8), 
where w, y,z are the equilibrium coordinates of any particle, p the density 
in equilibrium, u,v, w the displacements parallel to a, y, z, and » the 
function in question. @ is in fact the function the variation of which in 
passing from one state of the medium to another, when multiplied by da dy dz, 
expresses the work given out by the portion of the medium occupying in 
equilibrium the elementary parallelepiped da dy dz, in passing from the first 
state to the second. The portion of the medium which in the state of equili- 
brium occupied the elementary parallelepiped becomes in the changed state an 
oblique-angled parallelepiped, whose edges may be represented by dx (1-+s,), 
dy (1+s,), dz (1+s,), and the cosines of the angles between the second and 
third, third and first, and first and second of these edges by a, 3, y, which in 
ease the disturbance be small will be small quantities only. It is manifest 
that the function @ must be independent of any linear or angular displacement 
of the element dx dy dz, and depend only on the change of form of the element, 
* Cambridge Philosophical Transactions, vol. vii. p. 120. 
+ “On the Reflexion and Refraction of Light,” Cambr. Phil. Trans. vol. vii. p. 1. 
Read Dec. 11, 1837. 
{ They are virtually given, though not actually written down at length. 
§ Comptes Rendus, tom. liii, p. 1105 (1861). 
