Operative at one or more Pobits of an Elastic Solid. 387 



In like iiuinner wo may tiiid the rotations ot', txr", •vj'^'^ 

 (lofined by 



'iy - 'f = -1^', ^ _ fl = 2^". ^ _ ''^ = 2^". . (10) 

 ay dz dz d,v ' d.c dij 



For from (3), (4), (oj we have 



V"-V +/rt;r^ + i/>--WZ7^/^/ = 0, . . . (11) 



VV^h^V'-i/>--WZ7^/./' ==(),. . . (12) 



V2^"'-h/cV" =0, . . . (13) 



whence t3-"' = 0, and 



Zi fZ /6^-'^-'-\ „ Z, (Z 6?-^^'-x 



Ther^e are the results given in my paper o£ 1871. 



The values of 8, tn-', ixr", tn-'" determine those of a, /3, 7. If 

 we take 



.= -^pL, s=f^, \=p^ + w, . (15) 



d.v dz djj dz dz- 



where 



and 



^=rJ4v' (^'> 



it is easy to verify that these forms give the correct values 

 to h, -57', -sj", 'sj'". As regards the dilatation, 



in which 



Tills reproduces (9). 



As regards the rotations, we see that x ^^^^ i^^t influence 

 them. In fact 



j^ dia ,,_ diu 



and these agree with (14). The solution expressed by (15), 

 (16), (17) is thus verified, and it applies whether the solid 

 be compressible or not. 



2 02 



'S7"'=0, '5T=-|^— , -37=: — 



