AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 11 



And by repeating this process, we find finally that 



(«! - a,) («! - a„_i) (a, - a„_2) ( ) (a, - a^) = 0, 



which shews that some one of the constants a^, a^, &c. must be equal to a^. Suppose that 



Oi = flj, and put m, + Mj = m' ; then we have m' instead of the first two terms of (1), and 



du^ du 



= Oi instead of the first two equations of (2), and therefore, just as before, we may 



da? dt 



shew that 



(a, - a„) (ai - a„_^) ( ) (a^ - a,) = 0. 



Therefore Oj must be equal to some one of the quantities 03, Oj, 05, &c. ; let it be «3, then 

 proceeding as before, we may shew that 01 = 04, and again, that 0^ = 0^, and so on. We 

 have therefore 



tti ~ 02 = ds = ^n- 



(4.) The equations (A), (5), (C) in the preceding articles, may be very readily obtained 

 from the integrals of the equations of motion in the case of plane-waves of polarized light. 

 For when the wave-surface is a plane and the light polarized, we have 



a = aw, j3 = bii, y = cm, 



where u =f(vt — px — qy — sx), and a, b, c, any constants. 



By differentiating these expressions with respect to .v, y, z, and t, observing that p, q, s 

 are now constants, we have immediately the equations {A). 



To obtain the equations (B) and (C), we must put q = 0, 6 = 0, and then we have 



--(sy-ri)"=<»->(S)" 



Now a^ + c' = (ap + csy + (as — cp)', 



du du da du dy du da du dy 



also op— - =«a -— = ■«-—, cs--—=v-—, as—- = v-—, <•? -r: = '" ^- ^ 

 dt dx dx dt dz dt dx dt dx 



V" Ida dyy (da dyV' 

 hence — =— + --^+— --Jl. 

 v^ \dx dzj \dz dx I 



da d^ 



Now for transverse vibrations, we have ap + cs = 0, or 1 = 0, and for normal, 



dx dx 



da d'y 



as — cp = 0, or -i = : hence the truth of the equations (B) and (C) is manifest. 



dx dx 



(5.) If any of the quantities p, q or s, be imaginary, (a case we shall have to consider 

 hereafter) the first method of proving the formulae {A), {B), (C), fails, but the second method 

 does not. In such a case we call the vibrations transversal when the condition ap + cs = 

 holds; and normal when the condition as—cp = holds; and it follows easily from the 

 equations of motion, (see Cambridge Transactions, Vol. vii. p. 4l6) that transversal and normal 

 waves, thus defined, are in general propagated with different velocities ; i. e. the constant v is 

 different for these two species of vibration. 



(().) It is important to observe that, in articles (1), (2), the wave is supposed to be pro- 

 pagated in the direction PP', i. e. from P towards P". If therefore p, q, s be positive 

 quantities, the motion of the disturbance along PP' tends to increase x, y, and «r ; if p be 

 negative it tends to diminish x, if q negative y, and if s negative z. 



B 2 



