59 



ordinary equations of motion are included in the single symbolica- 

 tion equation 



dt* Xdx^dy^ dz*f ^ 'Vdx * dy^ r dz)\dx + dy+te)' 



If we employ the notation Au'.u, and assume the symbol 3D to re- 

 present the operation 



d p d d 



dx dy dz' 



the equation of motion becomes 



— =B(A3D.3D)v + (A-B) + AJD.t; ; 



or, by using the notation Du'.u also, it may be put in the form 



^ = {A3DA3D.-B(D3D.)°~}*>. 



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 JDU is 

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

 direction. 



The symbols A3D.v and D3D.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. 

 D3D.v represents, in magnitude, the degree oilateral disarrangement 

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

 that displacement takes place. 



These two symbols may be found separately by the integration of 

 an equation of the form 



J g U _ c /d*U d^U <PU\ 

 dt* \dx* dy* dz 2 )' 



When the six conditions above alluded to are introduced, the 

 equation of motion for a crystallized medium becomes 



+ 



+D *-{( B ^- B »fM B *§- B,i S> 

 ( B |- B *S>}' 



where A x A 2 A 3 are the three coefficients of direct elasticity with 

 reference to the three axes of symmetry, and B x B,' B 2 B 2 ' B 3 B 3 ' the 

 six coefficients of lateral elasticity with reference to the same axes. 

 If the vibrations be transverse, this equation is reducible to the 

 form 



^ = -(D3D.) 5 («^ + ^ + c 2 S 7 ) 



= — (D3D.) 2 (a' 2 aAa + 6°-/3A/3 + c^yAy)v, 



