VIBRATORY MOTION OF AN ELASTIC MEDIUM. 



515 



Also AB.B =(-)-.(-).(- 



Hence the equation of motion becomes 

 d-v 



dir 



= B(A'B.lB)v + (A - B)'mATB.v (3). 



We may also put it in somewhat a different form by using the notation Du'.u; for 



(AlZ9.i9)u - 19 AB.u = - (Dmy-v*, 

 therefore (3) becomes 

 d-v 



df 



= fJBAB. -B(Dm.y-}v 



(4). 



9. The symbol J3 has a very remarkable meaning which we shall now proceed to explain. 



denotes the rate of variation when <v alone is varied, that is, the rate of variation in the 



die 



direction a. To indicate this, we shall employ the notation d instead of - — ; i. e., if U be any 



dx 



quantity which is a function of w y x, and which therefore varies when we pass from one point 



to another of the medium, then d^U denotes the rate of variation of U, when we pass from point 



to point in the direction a- 



Now this rate of variation may be affected, like an ordinary velocity, with a sign of direction ; 



and it may be resolved or compounded in the same manner, and by the same rules, as an ordinary 



velocity. 



Hence, we may see immediately the meaning of the expression 



mU, or ad,U + (id^U +yd^U; 



for ad^U is the rate of variation of U in the direction «, affected with its proper sign of direction a, 

 fid^U is the rate of variation in the direction /3, and yd^U in the direction y, each affected 

 with its proper sign of direction. Hence, compounding these rates of variation as if they were 

 ordinary velocities, it follows, that the symbolical sum 



ad^U + fidf^U + yd^U 



expresses, in magnitude and direction, the complete rate of variation of the quantity U. 



10.. We may shew this differently as follows. 



Let a^ (i^ 7, be any three direction units at right angles to each other ; then it is easy to 

 prove, that 



"■,<l'a,+ f^A.+ '^Ay. = "''a + ^'^» + 7«^y' - (•')• 



Let us now choose a^ /3^ y^ so that a^ shall be in the direction of the normal to the surface 



dU =0, 



at the point .v y z ; in other words, supposing U to denote some disturbance or displacement of 

 the medium, a^ is chosen so as to be perpendicular to the surface called the /)■«?«< of the wave, for 

 dU = is evidently the differential ecjuation of that surface. 



• For let u and u' be any two linen, and let u rcprcHcnt the 

 direction unit of t*'; then, if u'-r'a, and « = aa + 4^J + c-, , we have 

 Du'.u=r'{liy-cll), and.'. /)«'. (flu'.u) = r'" (-i/i- fy). 



Now Au'.u' =r'^,auA n'Aii' u~r'''(in; thcrcforu 



Du'. ( Du'. u) or (Du')' . u = m'Au'. u - ( An'. u')u. 



