AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 13 



that the interval between two contiguous particles is extremely small, compared with the length 

 of a wave) ; therefore, by the principle stated in Article 8, the same must be true of the displace- 

 ments in the immediate vicinity of the plane of separation, which cannot be the case unless we 

 have a = a- Hence 



da . da da . 



Aa = -r— Ac» + — — Ay + -r-A« + &c 



dai dy dz 



Substituting these expressions for la, and Aa, and similar expressions for ^/3, ^7, A/3, A7, 

 and observing that all sums involving odd powers of Ix, Sy, Am or Ay, must vanish, in 

 consequence of the symmetrical arrangement of the system about the axis of z, but that sums 

 involving odd powers of Sz or Aaf do not vanish, since the particles are not arranged symme- 

 trically with respect to the plane of xz, we have 



X = - (C + D) — - D — + (C' + D') — + B' -^ + higher differential coefficients, 

 dz dx dz dx 



where - C = 2m/(r) Iz, - 2> = 2»i -/W ^Jo'lz, 



T 



C = 2W d)(r') Aar, D' = 2'w' i d>'{r') Aa?' A«. 



r 



(We assume the two first constants in a negative form, because Iz is negative, whereas A« is 



positive). 



It is evident that — C + C = 0, is one of the conditions of previous equilibrium, therefore 

 we have C' = C in the expression for X. 



Now since the length of the wave is extremely large compared with the sphere of action 

 of the molecular forces, the part of JC involving first differential coefficients, has its several 

 terms extremely large compared with those of the part involving second and higher differential 

 coefficients (see Cambridge Transactions, Vol. vii. p. 408) : therefore, unless the former terms 

 mutually destroy each other, JC will be extremely large compared with the corresponding force 

 which acts upon a particle at a distance from the plane of separation (for this force involves 

 only second and higher differential coefficients, see Cambridge Transactions, Vol. vii. p. 408) ; 

 and if this be the case, the vibratory motion of the particles at the plane of separation will 

 be extremely large compared with that at a distance from it, contrary to the ' principle stated 

 in Article (8). Hence the terms of X involving first differential coefficients must destroy each 

 other, and we therefore have 



(C+Z))^ + Z>^=(C+Z>')^-fi)'^' (1). 



dz dx dz dx 



In exactly the same way we may shew that /3' = /3, and 



(C + D)^ + i>^ = (C + i>')^' + i>'^ (2). 



dz dy dz dy 



Lastly, the force parallel to the axis of z is 

 -^.m {f{r)ly + - f{r)lz{lxla + lyl^ + lzly)\ 



+ 2W|^(r') A7 + -,<p'ir') A«(Aa7 Aa+ AyA/?+ A« A7)}, 

 which, treated as above, gives y = y, and 



(c,£)l^^z>(^ + ^)=(c + £:')^\z)'(^.f) (3). 



dz \dx dyl dz \dx dy I 



