Imperfectly Homogeneous Elastic Solid. 257 



Now equations (10) may be satisfied by u, v, w =w > v 0J w . sin £, 

 9 



where f = — - Qx + my + nz— Yt). If we find by differentia- 



tion -G7 l7 vr 2) w 3 , it is easily seen that they form a vector at 

 right angles to w r w , and to Z m n. If therefore its com- 

 ponents be U , V , W . cos f, we shall have 



JU +mV + ttW =0, "1 



£)U =-& , 



(V 2 -OU =-& 



(V 2 -^)V =-^ C 



(21) 



(V 2 -^)W =-^ , 

 where 



d «ajnJ +ftJf»V +'.^»V . 



The two values of V are therefore the reciprocals of the 

 semiaxes of the section of a 2 x 2 + b 2 y 2 + c 2 z 2 = 1 by the plane 

 lx + my + nz = 0, and the corresponding resultant rotations are 

 in the directions of these axes respectively. It follows from 

 this that if Z 1? m x , n x be the direction-cosines of the semiaxis VfS 

 and if from equations (21) we determine the value of 



V 2 (Z 1 U +m 1 V + rai W ), 

 this can be nothing else than 



V?ftU + m 1 V + n 1 W ), 

 whence the following Lem?na, 



(i»U — »„)«! + Q>*V -mS ) ni + (e?W - fl»)m 



= V?a i U + m 1 V + «,W ); . (22) 



which may be also verified directly with the greatest ease, 

 remembering that ll x + mm x + nn x = 0, and that 



, Z m n 



13. We turn now to the equations (18), by differentiating 

 which we obtain 



-*^-£+»'C 



eft 2 ~* v x dx * *^ \dy dz 

 PH. >%. S. 5. Vol. 3. No. 18. April 1877. 



