154 Prof. H. Hilton on the 



law for compounding angular velocities, that also 



0=0 a -0 l9 = 02-01, ^=^2-^!, 



provided all the quantities concerned are small. 



Hence the work done by the mutual reaction of Mi and M 2 

 in the small displacement of both is a linear function of 

 (f 2 — ?i) 3 j (02""^i) 2 > & c -> terms of the first degree in £ 2 — ?i> 

 &c, being left out of consideration. 



§ 6. Consider, now, the orthorhombic crystal described 

 in § 1. We shall suppose the crystal isolated in space. An 

 alternative is to suppose the molecules in the planes 



df=l, £~l, z/ = l, i/ = m, 2=1, z = n, 



i, e. in the faces of the crystal, all fixed. The reader will be 

 able to write down the solution in this case, taking § 2 to 

 guide him instead of § 2>, as is done below. 



Suppose the centroid of the molecule whose equilibrium 

 position is (pa, qb, re) is given a small displacement to the 

 position 



(j9a-K%5,n qb + y p , q ,r, rc + z p , q>r ), 



while the molecule receives a small rotation through an 

 angle e about a line through its centroid whose direction- 

 cosines are 



J cosec J e P> q> r , -J cosec | e <f> Pt q> r , J cosec -J e -^ 9 , n 



as in § 5. Then the whole work done on the crystal in such 

 displacements is small of the second order, for the displace- 

 ments take place from a position of equilibrium ; and by § 5 

 the work done is a linear function of the squares of quantities 

 such as 



®p,q ~~£p,q,r — l) Up,q,r yp,q,r-\t *p, q, r 2p,q,r-li 



Vp,q,r — ^p,q,r-h TP>^ r — rP' <2> > — *9 TP><i' r rP'5'''- 1 ' 



and does not involve their products. 



It will follow, as in §§ 2, 3, that the small translational 

 oscillations parallel to the axes of reference and the three 

 rotational oscillations given by p , q , r , <l>p,q,r> tyv,q,r are a ^ 

 independent and are obtained from equations of the same 

 type. 



it will be sufficient, therefore, to consider translational 

 oscillations parallel to the z axis. The equations of § 3 (ii.) 



