Cambridge Philosophical Society. 379 



equations derived from this hypothesis be true, a crystallized medium 

 is incapable of propagating transverse vibrations. 



Secondly, respecting the use of the symbolical method and notation 

 above alluded to. 



The application of the symbolical method and notation to the subject 

 of vibratory motion is very remarkable, and leads to equations of 

 great simplicity. In the case of an uncrystallized medium, the three 

 ordinary equations of motion are included in the single symbolica- 

 tion equation 



dt^ \d9!'' dy'' dz^i ^ \ dx '^ dy ' dz ) \dx ^ dy ^ dz ) 



If we employ the notation Am'.m, £md 9,ssurae the symbol 50 to re- 

 present the operation 



d I /oj^ , d 

 dx dy dz ' 

 the equation of nogtion becomes 



^ =B(AlD.3D)v+(A-B)-i-AlD.v ; 



or, by using the notation Dm'.w also, it may be put in the form 

 ^ = { AIDAlD.-B(DlD.)°-}y. 



The symbol 3D written before any quantity U which is a function 

 of xyz, has a very remarkable signification ; the direction unit of the 

 symbol 3DU is that direction perpendicular to which there is no va- 

 riation of U at the point xyz, and the numerical magnitude of 3DU is 

 the rate of variation of U, when we pass from point to point in that 

 direction. 



The symbols AiP.u and DiD.v have also remarkable significations. 

 AlD.u is a numerical quantity representing the degree of expansion, 

 or what is called the rarefaction of the medium at the point xyz. 

 DtD.u represents, in magnitude, the degree oi lateral disarrangement 

 of the medium at the point xyz, and, in direction, the axis ^bout which 

 that displacement takes place. 



These two symbols may be found separately by the integration of 

 an equation of the form 



d''\J _^ (d^V rfnj rf2U\ 

 dt^- \c?j?2 dy'^ dz^J' 



When the six conditions above alluded to are introduced, the 

 equation of motion for a crystallized medium becomes 



