Solid Cylinders of Elliptic Section. 77 



validity. Here attention will be directed only to the stress- 

 difference and greatest-strain theories*. 



In applying these theories we may confine our attention 

 in the first place to the first approximation. Doing so, it 

 may be proved by somewhat laborious analysis, omitted here 

 as in itself of no interest, that the greatest values both of the 

 stress-difference and greatest strain occur at the centre of the 



disk. They are respectively the values of sex and ~- for 



x =y = 0. Denoting the maximum stress-difference and great- 

 est strain by S and "s respectively, we find 



S=o>V 2 K, (23) 



i =o>V 2 (l-i?6 8 /a 2 )-K/B = {1-fjP/cP) S/E, . . (24) 



where K is given by (20). 



§ 7. We may attach to the limiting speed at least two 

 different values. Taking, for instance, the stress-difference 

 theory, and measuring force in tons weight and length in 

 inches, we may regard S as the greatest longitudinal traction 

 in tons weight per square inch under which the material in 

 question, in the form of a bar, satisfies Hooke's law " stress 

 proportional to strain " with sufficient exactness for the 

 legitimate application of the mathematical theory. We may, 

 however, regard S as the number of tons weight per sq. 

 inch which engineers consider the safe working limit in the 

 material, provided this be taken sufficiently low to satisfy the 

 linearity of the stress-strain relations. The former view 

 would unquestionably be theoretically the more satisfactory 

 if experiment showed a distinct point in stress-strain diagrams 

 where Hooke's law ceases to hold. Even, however, if such a 

 point did exist when the load on the bar was gradually raised 

 in a particular way, it might be rash to assume that the same 

 definite point would present itself in the material of a rotating 

 cylinder whose speed was being gradually increased. 



An objection to the second view is the wholly arbitrary 

 nature of the limit it assigns to S, and the fact that the safe 

 load varies with the engineer, and is likewise an unknown 

 function of the contingencies to which he surmises the parti- 

 cular structure may be exposed. It thus appears best to 

 present the results in such a way that a reader may attach his 

 own values to S or 5, and may have a minimum of trouble in 

 deducing numerical results. 



* See Phil. Mag. September, 1891, pp. 240-241. 



