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



produce the same terms in the differential equations of motion, as X^, Y-, Z-; 

 for the squares will be the same in each, and the rectangles will be 



(2/3372-4^3273, 27103 -4a,73, 2a,/3, - 4a,^o), and (-2/337.,, -27103, -2o,/3i). 



These two sets of rectangles will produce the same terms in the equations of mo- 

 tion, since we may transpose the differentiations without affecting the result, so 

 far as the laws of propagation are concerned. The surfaces of wave-slowness de- 

 duced from (38) and (39) may be obtained immediately from equation (18). 

 Equation (38) will give the following: 



P ^AF + Em' + Qn' ; F' = {A - P) mn ; 



Q' = Am' + Pn' ^ RP ; G' ^ {A -Q)ln; (40) 



E ^ An' -f Ql' + Pm'; H' = (^1 - R) Im. 



And similarly from equations (39) wiU be found 



F = Em' + Qn'; F' = - Pmn ; 



Q' = Pn' + El'; G'=- Qln ; (41) 



E=Ql' +Pm'; H' = -Elm. 



These equations differ from the former only by not containing A. 

 The equation of wave-slowness (18) derived from (40) is 



\A{,T' + f+z')-l\ 

 X ! Or+f+s')(QE.v'+PEy'+PQz')-(Q+Ey-(P+E)y'-(P+Q)z'+l \=0. 



(42) 



The first factor of this equation represents a sphere whose radius is — rj, and 



belongs to the normal vibration ; the second factor is the equation of Feesnel's 

 wave-surface, and in it the vibrations are transversal. Tlie equation of wave- 

 slowness deduced from Professor Mac Cullagh's function (39) will be the last 

 factor of (42). So far, therefore, as the laws of wave-propagation are concerned, 

 the functions (38) and (39) are eqiiivalent, with this difference, that the func- 

 tion (38) introduces a normal wave, which does not enter into the equations 

 derived from ( 39). It might be thought at first sight that we are at liberty to 

 make .4=0, and thus reduce the function (38) to a function representing 

 jiothing but transverse vibrations ; this, however, cannot be admitted, for as 



