74 Mr. C. Chree on Rotating Elastic 



Fdz=0, Gtdz=0, (11) 



instead of as previously from F = 0, G = 0, when z=0. 



§ 3. We may spare ourselves the trouble of redetermining 

 the constants, as the requisite changes are easily hit on and 

 still more easily verified. In the formulas (125), (126), and 

 (128) of my previous paper for a, /3, and A, we have only to 

 replace z 2 by z 2 — l 2 /3, and in the formula (127) for 7 we 

 now write z (z 2 — ! 2 ) for z d . The verification is easy if it be 

 noticed that this adds to the previous values of a, /3, 7, and A 

 terms Pafa 9 l 2 fi'y, l 2 y f z, and I 2 A' respectively, where a', /3', 7', 

 A' are constants satisfying 



a' + /3' + 7 ' = A'. 



The internal equations (3), above, contain only second 

 differential coefficients of the displacements, and so are 

 unaffected. Also, it is easily proved that in our new solution 

 we have everywhere 



yz=za= zz=0 (12) 



Thus the conditions (4), (5), (6), and (9) are identically 

 satisfied, and, lastly, it is easily verified that the new solution 

 satisfies (11). The solution is the following : — 



EA(3a^ + 2a 2 b 2 + 3b i )^{co 2 p(l-2 V )} = (a 2 + b 2 )\a* + a 2 b 2 {l+r ) )+b i } 

 -a? {a* + a 2 b 2 + W + rjb\o? + 3b 2 ) } -tf \ 2a* + a 2 b 2 + 1/ + V a 2 (Sa 2 + b 2 )} 



+ v( i* V h il 2 -z 2 ){3a i + 2a 2 b 2 + 3b*), (13) 



E« (3a 4 -f 2a 2 b 2 + SV) /co 2 p = x(a 2 - V b 2 ) {a 4 + a 2 b\l + rj) + b"\ 



+ (Ll 2 -z 2 )xr } {a* + a 2 b 2 + 2b 4 + r ) b 2 (a 2 + 3b 2 )}, (14) 



E/3(3a 4 + 2a 2 b 2 + 3h±)/co 2 p=y(b 2 - V a: 2 ){a i + a 2 b 2 (l + V ) + b*\ 



+ {±l 2 -z 2 )yr ) {2a 4 + a 2 b 2 + b 4 + r ] a 2 (3a 2 + b*)}, (15) 



E r (3a 4 + 2a 2 6 2 + Sb 4 )/a> 2 p =-z V (a 2 + b 2 ) { a 4 + a 2 6 2 (l + v) + #} 



+ zx? V {a 4 + a 2 b 2 + 2b 4 + rjb*{a* + 36 2 ) } + zf V { 2a 4 + a 2 b 2 + b 4 + *?a 2 (3a 2 + b 2 ) \ 



^.L Z Q2_ Z 2^ V 2 (1+V) ( 3a 4 + 2 a 2 6 2 + 36 4 ) v . . (16) 



