288 Mr Ghree, On Long Rotating Circular Cylinders. [Feb. 8, 



The numerical results in these two tables are exact*. Both ha fa 

 and ha' ja' are linear in rj ; thus results for other values of w are 

 easily and accurately supplied by interpolation. The results 

 under a' fa = apply also to the solid cylinder in Table II., but 

 not of course in Table III. The formulae (10) and (11) are 

 identical with (45) and (46), Proceedings, I.e. p. 213, which give 

 ha and ha' for a thin disk. Thus Tables II. and III. apply also to 

 thin disks. 



The large and steady increase in the value of ha/a as a'/a 

 increases is very conspicuous. It is also noteworthy that when 

 a'/a has a given value, ha' /a increases but ha/a diminishes as 77 

 increases. In fact by (10) and (11) 



ha/a -h ha' /a = (to a P a*/E) (1 + a' 2 /a 2 ), 



so that ha/a + ha' /a' is independent of 77. When a'/a approaches 1 

 the influence of r\ on the magnitude of the alterations in the radii 

 tends to disappear. An idea of the numerical magnitude of 

 ha/a and ha' /a' will be most easily derived from the special cases 

 treated in Tables IX. and XI. 



For the alteration ha — ha in the wall thickness we have 

 the same formula as for a thin disk, viz. (47), p. 213, and this 

 thickness is increased or diminished by rotation according as 



a'/a < or > (1 — Jn) + (1 + Jn) ; 



see Proceedings, I.e. Equation (48), p. 213, and subsequent remarks. 



Comparing Tables I., II. and III. it will be seen that ( — hl/l) 

 is by no means negligible compared to ha/a and ha' fa unless 77 

 be small. Thus as afl is small, the shortening of the cylinder 

 should in general be more easily detected than the alterations 

 in its radii. 



§ 9. Since ™ is zero the principal strains are everywhere 



dw du -, u 

 -v— , -7- and - . 

 dz dr r 



* This term as applied to this and following tables means that the numerical 

 results are as exact as the formulae, and not merely the first figures of a decimal. 



