114 The Rev. S. Haughton on a Classification of Elastic Media, 



limits will be M.' -t- c^dydz + LB -r- l^dxdz ; which cannot be reduced to the 



former value, unless A' = B. 



It is sufficient here to establish the general theorem. I shall return to it 

 again, in comparing Professor Mac Cullagh's theory of light and that ad- 

 vocated by M. Cauchy and Mr. Green, which give the same laws of wave- 

 propagation, but differ in the laws of reflexion and refraction. 



I shall now integrate the equations of motion (14), for the particular case 

 of plane waves and rectilinear vibrations. Let (/, m, n) be the direction cosines 

 of the wave-normal, (a, /3, 7) the direction of molecular vibration, and {v) the 

 velocity of the wave ; then the integral will be 



^ = cos a-f\ — {h -f my + nz ~ vt) i, 



»; = cos j3 •/ 1 -£^ {Ir. + my + nz ~ vt) |, 



f = cos 7 •/ 1 ^ {Ix + my + nz - vt) j. 



Introducing these values into the equations (14), we obtain 



eV^ COS a= P' COS a + H' COS ji + G' COS 7, 



ev- COS j3 = Q' COS /3 -t- F' cos y + H' cos a, (16) 



ev- cos y = R' cos y + G' cos a + F' cos /3 ; 

 where 



P' = («,=) i- + (a/) m- + (as") n" + 2(a,«3) 77m + 2(a,a3) In + 2(a,«2^ Im ; 

 Q' = (b,') P + (5/) m' + (is-) n' + 2{bJ),) mn + 2(bA) In + 2(bA) Im; 

 R' = (c,2) f + (C2-) m^ -t- (03-) ri' + 2{CiCi) mn + 2{c,C:,) In + 2{c,c^) Im; 

 F' = {btCi)P + {b2C2)m^+{b3C3)n^+{b2C3+b3Ci)mn+{biC3+bsCi)ln+{b,c.2+b-,Ct)lm ; 

 G'= {atC,)l%(a2C2)m-+(a3C3)7i-+(a2Ca+a3C2)inn+{aiC3+a3Ci)ln+{aiC.,+U2Ci)lm; 

 H'= {aA)l'+{a2b2)m-+{a3b3)n^+(a2b3+a3b2)mn+{aib3+a3l>i)ln+(aA+ail'i)lm. 



Equations (16) are the well-known equations for determining the axes of 

 the ellipsoid whose equation is 



Fw- + Q!y^ -f Bz' -f -iF'yz -f 2 G'xz + 2H'xy = 1. (17) 



