Fresnel's Theory of Double Refraction, 26 



are but demonstrations, in perhaps a somewhat new form, of 

 one or two points of Fresnel's Theory of Double Refractions. 

 Such as they are, however, I submit them to your notice, as 

 obvious and satisfactory deductions from the vahial)le papers 

 on the undulatory theory which have recently appeared in your 

 periodical. 



On reference to Mr. Tovey's communication in the January 

 number, 1836, (vol. viii. p. 8,) it will be seen that 



If wi, Wi, 7??2, &c. be the molecules of an elastic medium, 



ar, y, 2, the coordinates of rest of tw, 



X -\-hpy -^k^,z ^-l^ ofm^ 



&c 



y^ = Vhf + kf + If, &C. = &C., 



and if, the system being disturbed, the displacements of m at 

 the time / be z/, x;, w || the coordinate axes, those of m^ 

 u + ^fU, V -i- ^fV, w + dfW, &c., then the accelerating forces 

 on m in consequence of the disturbance will be 



^m {<pr .^u + {h^u -\- kdv + nw)/l^lfr} || ^, 



^m {^r,'8v -{■ (hdu + k^v -\- Idw) k^r} \\y, 



Sm{(pr,^w-]-{h^u + khv + n'w)l^i;r} \\ z, 



f r f^T f r 



where 4> r =^-—i 4/ r = "^— ^ — "^-j, the action of a molecule 



771 on another at the distance r from it in the direction of their 

 distance being m f r, and the sign f extending to all the 

 molecules within the sphere of m's action. 

 Now at the beginning of the motion we have 



d,u = ^^u= &c. = — M, 



8^ v = §2 w = &c. = — t;, 



S^tt) = §2^ = &C. = — -KJ, 



m„ mg, &c. not having yet been disturbed ; and . • . 

 initial force \\x = Au -\- Hv + Gw = p 



{A cos a + Hcos^ + Gcosy}, 



\\^ ^iUu + Bv -\- Fw = p 



(Acosa+Bcos/3 + Fcosy}, UA,) 



\\z=:Gu + Fv-i-Cw = p 



{ G COS a + F cos /3 + C cos y I 



where p is the whole displacement of the molecule, making 

 4r s a, /3, y with the axes, and 



Third Series. Vol.10. No. 58. Jan, 1837. E 



