156 Mr. C. Chree on Rotating Elastic 



E/3(l -rj) (3a 4 4- 2a 2 6 2 + 2>V)/a>*p = 

 y {b* (a 4 + a%* + V) - V (a 6 + 2a 4 6 2 + a 2 6 4 + 6 6 ) + £^(a* - ^) (3« 4 + V) 



+ lv 3 (a*-b*f(a* + b*)\ 

 -i{l + v)y^a i + a%* + b*- V (3a' i + a%* + 2b 4 )- V z(a*-b i )} 

 -(l+ V )yx*{b±- V a*(a 2 + b*) + v *(a*-b*)}, (78) 



7 =_o> 2 ^(a 2 + 6 2 >-r(4E), (79) 



£*(1 -17) (3a 4 -f 2a 2 6 2 + 36 4 )/a>V = (a 2 -^ 2 ){a 4 + a 2 6 2 + ^-^(a 4 + 6 4 ) } 



-y*a\l + 2 V ), . . . (80) 



^(1 -17) (3a 4 + 2a 2 6 2 + W)/<o*p = (6 2 - */ 2 ) {a 4 + a 2 6 2 + 6 4 - 9/(a 4 + V) } 



-x%\l + 2 v ), . . . (81) 



^(l-i7)(3a 4 + 2a 2 ^ 2 + 36 4 )/a)V = 477ra 2 4^){(a 2 + 6 2 ) 2 -97(a 2 --^ 2 ; 2 } 

 .-^ 2 {a 4 + a 2 6 2 + 26 4 -7 7 (a 4 -6 4 )}-^ 2 {2a 4 + a^ 2 + 6 4 + 7 ? (a 4 -6 4 )},(82) 



^(l-97)(3a 4 + 2a 2 6 2 + 35 4 )/ft) 2 /3 =-.^{a 4 4-6 4 -7 ? (a 2 + ^) 2 }, . . (83) 



yz = zx = ^84) 



§ 36. As already stated, zz vanishes when rj = 0. For 

 other values of rj the solution applies only under the same 

 restrictions as Saint- Venant's solution for beams, and portions 

 of the cylinder immediately adjacent to the ends should be 

 excluded from its domain. 



At every point zz is one of the three principal stresses, 

 but the other two vary in direction from point to point of 

 the cross section, being parallel to the axes of the ellipse only 

 at points which lie on these axes. The stress system other 

 than zz may be conveniently analysed into a series of simple 

 systems. For shortness let 



{a 4 + a 2 6 2 + b*- V {a* + Z> 4 )} -r- {(1 - V ) (3a 4 + 2a 2 6 2 + 3/> 4 )} = K', (85) 



then the simple systems are as follows : — 



(i.) A uniform normal tension a> 9 pa*'K / parallel to the 



major axis, 

 (ii.) A uniform normal tension (D^pb^EJ parallel to the 



minor axis. 



