ON DOUBLE REFRACTION. 277 
medium in its undisturbed state to be free from pressure. The tensions 
P, Q, R, P,, &c. will now be small quantities of the first order, so that in the 
formule (22) and (23) we may suppose the tensions referred to a unit of 
surface in the actual or the undisturbed state of the medium indifferently, 
and may moreover in these formule, and in the expression for ¢, take a, y, z 
for the actual or the original coordinates of a particle. 
Green assumes as self-evident that the value of g for any element, suppose 
that which originally occupied the rectangular parallelepiped dw dy dz, must 
depend only on the change of form of the element, and not on any mere 
change of position in space. Any displacement which varies continuously 
from point to point must change an elementary rectangular parallelepiped 
into one which is oblique-angled, and the change of form is expressed by the 
ratios of the lengths of the edges to the original lengths, and by the angles 
which the edges make with one another or by their cosines. If the medium 
were originally in a state of constraint, @ would contain terms of the first 
order, and the expressions for the extensions of the edges and the cosines of 
the angles would be wanted to the second order, but when ¢ is wholly of the 
second order, those quantities need only be found to the first order. It is easy 
to see that to this order the extensions are expressed by 
du dv dw 
da dy’ qat e Beet ail NESS (24) 
and the cosines of the inclinations of the edges two and two by 
dv , dw dw , du du , dv 
dat dy’ ag ae ype 7402 ay 3 (25) 
and @ being a function of these six quantities, we have from (23) 
T,-=2.,, Tee=T;-5 T y= Ty. ei <i Re eta CED 
These are the relations pointed out by Caucky between the nine components 
of the three tensions in three rectangular directions, whereby they are reduced 
to six. The necessity of these relations is admitted by most mathematicians. 
Conversely, if we start with Cauchy’s three relations (26), we have from (23) 
(dw [dv (le _ »fdw [QW _ {du rs 
Ma-(z) ee) tG)=FG@). © 
The integration of the first of these partial differential equations gives 
f=a function of we and of the seven other differential coefficients. 
Substituting in the second of equations (27) and integrating, and substituting 
the result in the third and integrating again, we readily find 
f=a function of the six quantities (24) and (25). 
We see then that Green’s axiom that the function » depends only on the 
change of form of the element, and Cauchy’s relations (26), are but different 
ways of expressing the same condition ; so that either follows if the truth of 
the other be admitted. 
Cauchy’s equations were proved by applying the statical equations of 
moments of a rigid body to an elementary parallelepiped of the medium, and 
taking the limit when the dimensions of the element vanish. The demonstra- 
tion is just the same whether the medium be at rest or in motion, since in 
the latter case we have merely to apply d’Alembert’s principle. It need 
hardly be remarked that the employment of equations of equilibrium of 
a rigid body in the demonstration by no means limits the truth of the 
theorem to rigid bodies; for the equations of equilibrium of a rigid body are 
