Mr. Moon on Fresnel's Theory of Double Refraction. 135 



respectively are not the same as if we calculate the effect of a 

 disturbance communicated in the direction of the greatest, 

 and then of another communicated in the direction of the least 

 diameter. In the latter case, if u and v be the co-ordinates 

 of the particle in the plane of the section, respectively parallel 

 and perpendicular to the greatest diameter, the equation of 

 motion parallel to that line is 



d*u 



dt 



-Aw (1.) 



In the former, if a/3y be the inclinations of the greatest dia- 

 meter to the axes of elasticity, we have 



(Pu 



-7- ^ = a 2 .r cos a + ftycosfi + c 2 zcosy; . . (2.) 



but if r be the radius vector of the particle, 



u = r cos 5 = x cos a. + y cos /3 + z cos y, 



from which it is obvious that equations (1.) and (2.) can never 

 be identical. This single circumstance would alone be suffi- 

 cient to condemn the whole theory; I mention it however 

 chiefly to show the gross fallacies which have been unhesita- 

 tingly received into it. 



As to the second point which I have asserted, that ho dis- 

 turbance whatever will be propagated from the originally dis- 

 turbed particle, a circumstance which if true must scatter the 

 whole theory to the winds, I must say I approach the discus- 

 sion of it with considerable pain, when I reflect that a result 

 so immediately and incontrovertibly flowing from Fresnel's 

 assumptions should so long have been overlooked or disre- 

 garded ; and this when the theory has for years been subjected 

 to the scrutiny of the ablest philosophers of this and of other 

 countries. 



Assuming Fresnel's proof of the axes of elasticity to be 

 genuine, we get the following equations for determining the 

 motion of the disturbed particle: — 



&x _ 2 

 dt*~~ a ?? 



tl- _ 2 

 dt*~ ° *' 



