and of the Axes of Elasticity, in Biaxal Crystals. 295 



that of Fresnel, respecting the mechanical signification of the 

 axes of elasticity. The existence of three rectangular axes 

 possessing peculiar properties is not a principle, but a result, 

 of theory ; their directions are determined by conditions per- 

 fectly analogous to those which determine the principal axes 

 of an ellipsoid from its general equation ; and these directions 

 are functions of certain quantities which are constant when 

 differentials of the second and subsequent orders are neg- 

 lected, but which vary when these are taken into account. The 

 differentials of higher orders introduce terms depending on 

 the wave-length ; and thus the directions, as well as the lengths, 

 of the principal lines depend on the colour of the light, or, to 

 speak more accurately, on the length of the wave. 



All this will be easily understood if we recur to the first 

 principles of the theory. According to these, everything de- 

 pends on the form assigned to the function V in the general 

 dynamical equation 



Jffi^d^ i+ d ^+ d ^n)=fffd^yMy, 



from which the motion of the aether is deduced. In my first 

 memoir on the subject (read to the Academy on the 9th of 

 December, 1839), I showed that when differentials of the 

 first order only are preserved, the function V — which may 

 perhaps with propriety be called the potential, since the mo- 

 tion of the system is potentially, or virtually, included in it — 

 is a function of the second degree, composed of the three 

 quantities X, Y, Z, which are connected with the displace- 

 ments £, ij, f by the following relations : — 



x _d_>?__£?jr y=— — — Z — ~ — — 



~~ dz dy 9 ~ dx dz ~ dy dx' 



To show this, I make use simply of the consideration that the 

 motion must be such as to satisfy the condition 



d% dr) d£ „ 

 dx dy dz ' 



which seems to be characteristic of the vibrations of light. 

 But the same condition allows us to suppose that the poten- 

 tial contains not only the quantities X, Y, Z, but their dif- 

 ferential coefficients of any order with respect to the coordi- 

 nates. This supposition, however, is too general, and re- 

 quires to be limited by other considerations. Now the most 

 natural restriction which can be imposed consists in the as- 

 sumption that the quantities of all orders are formed on the 

 same type, those of any order being derived from the prece- 



