158 Mr. C. Chree on Rotating Elastic 



smaller eccentricity than the surface of the rotating cylinder. 

 It is also easy to prove 



a n < a. 



Thus the surface of no longitudinal stress lies wholly within 

 the cylinder when 



hi < b, 



L e. 2&V + ^) 2 - (a 2 ^^) 3 +i7(a 4 -5*)(a 9 +3& 2 ) >0. 



When this inequality holds the portion of the cross section 

 wherein the longitudinal stress is a pressure forms a complete 

 annulus limited externally by the surface of the cylinder and 

 internally by (88). When, however, the above inequality 

 does not hold — and by taking bja small enough it can always 

 be reversed even when rj = 'b — the portion of the cross 



section wherein zz is a pressure consists of two detached areas 

 surrounding the ends of the major axis. 



§ 38. When rf-0 the maximum stress-difference is always 



correctly given by the axial value of xx. But for other values 



of 7] it is given by the axial value of xx-yy or by the axial 



value of xx-zz according as bja is less or greater than a 

 certain value. This value of bja increases with rj, being 

 approximately '217 when r/ = *25, and *511 when rj=. '5. The 

 greatest strain is always correctly given by the axial value of 



da. — 



— . The expressions for the maximum stress-difference S 

 ax 



and greatest strain s may easily be found from equations 

 (77)-(82). It seems unnecessary to write them down. 



The limiting safe speed cannot be determined solely by 

 reference to a limiting elastic stress or strain on account of a 

 species of instability which may arise. This question of 

 instability will be discussed presently, but in the meantime it 

 is convenient to record results from which the limiting speed, 

 according to the stress-difference and greatest-strain theories, 

 might be simply derived when the circumstances are such 

 that these theories apply. In Table XIII. the angular 

 velocity is termed w 1? and in Table XIV. it is for the sake of 

 distinction termed o) 2 . Assigning to S and i in these tables 

 their limiting values for the material under consideration, we 

 obtain the limiting speeds according to the stress-difference 

 and greatest-strain theories, while by assigning a given value 

 to (o x a and co 2 a we obtain the corresponding maximum stress- 

 difference and greatest strain. The tables should be compared 

 with Tables I. and II. 



