556 Prof. A. M c Aulay on 



The potential energy W for the whole of space is now 

 given as the volume integral of w m or of w thus 



W = §yw m d<s=$wd<Si (7) 



where we have the following for w m 



(MacCullagh form) w m = -&(YVv)WV*) +iK^V l 0% (8) 



and the following for w 



(Green form) w = S(V65)A(V^5 1 +&) + i&(S£K) 2 - . (0) 



By transforming 



8(V&6)a(V x 6xW 



by putting ^ = -^ + Y\( ) it is easy to show that 



= -8VlV2V3Sll1 2 V35 , 3-> 



(10) 



and therefore that the volume integral of w—w m for the 

 whole of space is zero. 



It will be found that with h constant for a crystalline 

 volume, (9) is Green's 1839 form*. When b is zero and h 

 constant, (8) is obviously MacCallaglr's 1839 formf. 



To show that the equilibrium position of the solid is stable 

 it is necessary only to show that io m o£ (8) is positive. With 

 the specification of § 1 we may, I think, regard this as obvious. 

 We may take any point inside a crystal, say its centre of 

 volume, and leave that point with the same intrinsic dis- 

 placement as for free aether. The interface may be assumed 

 to be of uniform thickness, say 10 — 6 cm., and in the three 

 principal directions of the crystal e^ f\, g ± may be supposed, 

 through the interface, to change continuously to e =f =g ' 



The theory that we have been thus led to differs in form 

 from MacCullagh's in that for stability we are not permitted 

 to put b = 0; it really differs from Green's in the interface 

 conditions. It may be said further to differ in form from 

 MacCullagh's in that h is large for the interface region as 

 compared with the rest of space, and therefore perhaps we 

 shall find that when we treat the interface as a surface of 

 discontinue we must reckon with a surface contribution to 

 the potential energy. 



* " On the Propagation of Light in Crystallized Media/' Math. Papers, 

 p. 293, Trans. Camb. Phil. Soc. 1839. 



+ An Essay towards a Dynamical Theory of Crystalline Eeflexion and 

 Refraction. Collected Works, p. 145, Trans. Roy. Ir. Ac. 1639. 



