in the Theory of Double Refraction. S53 



necessary to show that the coordinate axes may always be so 

 assumed that the products 



m% {-^ (r) At/ a ^ a ij} &c. 



are equal to zero. 



m% {^{r)Aj/AxAri} = 



— 2mA cos Ycos {n t—k r) S {^^ [r) sin^ (~s~~) ^^ Ay} 



+ mAcos Y^m{nt ^ kr)^ { ^ (^O sin(^ Ar) A.r A^}. 

 If the medium is constituted so that the molecules are dis- 

 tributed symmetrically, as supposed by M. Cauchy, " Si les 

 masses des molecules m, m\ m'', &c. sont deux a deux egales 

 entre elles, et distribuees symmetriquement de part et d'autre 

 d'une molecule quelconque m sur des droites menees par le 

 point avec lequel cette molecule coincide," {Nouveaiix Exer- 

 cises, p. 10.*) 



2 { « \I/ r sin {k A r) A x Aj/} = 



as is obvious without any proof, and it is only necessary to 

 show that 



S-rtI/rsin^(-|^') AxAyX = 



t |rl/rsin^(^) A^A;.| =0 (A.) 



S|"4'/-sin^(^') AyAzX =0 



Let X =. aoi} ■\- by' -\- c z' Ax =i a A j;' + Z> Aj/' + c A z' 



y = a'.r + <^V+ (^^ ^y = a'Ax' + b'Ay'+ c' A z ' 



z = a'' of + b"f + c"z' Az - a" A x' + V'Ay' + d'A z . 



a^ + b^' + c^ = I «'3+i'3 + c'^ = 1 



a"2 + ^// 2 ^ p// 2 ^ 1 a' a!' + b' W + 6-' c" = (C.) 



aa" + bb" + cc" = aa' ■\-bU ^ cd = 0. 



^^' s{^Kr)sin^(^|L!-)A..'^}=D 



s{rKr)sin^(^*)Ay^}=E 

 2 i" rl/ (r) sin« (-|- ) A 2'^ y = F 

 S <{" ^^ (r) sin2 [^^) Ay' AzfX = D' 



S / ^^ (r) sin^ (^^) ^^' ^ ^'T = E' 



.* Mr. Tovey makes use of the same assumption, Lend, and Edinb. Phil. 

 Mag., 1836, vol. viii. p. 10. and p. 270. This condition will not generally 

 hold at the confines of any medium. 



Phil. Mag. S, 3. Vol. 15, No. 97. Nov, 1839, 2 A 



