160 



Mr. C. Chree on Rotating Elastic 



The results are all exact. This table should be compared with 

 Table III. 



§ 40. In the cross section the most important displacements 

 are the alterations in the lengths of the principal axes. The 

 major axis always lengthens under rotation. The increase 

 ha in the semi-axis is got by putting x = a, y = in (77). 

 The alteration 8b in the minor semi-axis is got by putting 

 #=0, y = b in (78). 



When 77 =0 it is easily seen that the minor axis lengthens 

 under rotation for all finite values of b/a. For other values 

 of rj, however, the minor axis shortens when b/a is less than a 

 critical value bja. The value of by/a increases with 77 ; thus, 

 answering to t; = *25, '3, *5 we find approximately 



6 1 /a = *584, *663, *783 respectively. 



These do not differ very much from the corresponding results 

 in the case of a thin disk (see Table VI.). In the following 

 table of values of 8a/a and 8h/b no sign is attached to the 

 former quantity as being always positive. 



Table XVI. 

 Values of (8a/a)-*-(©Va a /E) and (S6/6)~(©V 2 /E). 



b/a= 



0. 



•2. 



•4. -6. -8. 10. 



(da/a)- 



(Sb/b)- 



-( W > 2 /E) = 



\V=0 

 n=-2o 



U = -5 



\ n =-2b 



•2 



•224 

 •229 







-•0885 

 -•1875 



•225 

 •226 

 •230 



+ •0090 

 -0794 

 -1780 



•233 -242 -248 -25 

 •229 -229 -216 -1875 

 •231 -220 -187 -125 



+•0372 +-0870 +-1587 4- -25 

 -•0496 +0052 +-0858 +-1875 

 -•1458 -0840 +0087 +'125 



The results for b/a = 1 are exact, the others are nearly all only 

 approximate. Tbis table should be compared with Table IV. 

 § 41. The expressions for the displacements and strains in 

 the long cylinder are more complicated than in the thin disk, 

 and their full consideration would require more analysis than 

 the interest of the results seems likely to warrant. I shall thus 

 merely call attention to the more striking features of the 

 radial strain along the principal axes of the cross section. 



Along the major axis the radial strain is the value 



with y = 0. In a circular section when t) —'5 the radial strain 



f dot 

 dx 



