Solid Cylinders of Elliptic Section. 171 



where k is the radius of gyration of the elliptic cross section 

 about a perpendicular to its plane through its centre. 



Again, let A = 7rah denote the area of the section of an 

 elliptic disk, and let 8A Z be the increase produced by rotation 

 in a section at distance z from the central plane. Then 

 denoting the direction cosines of the outwardly directed 

 normal to the rim by \ and fi, we have 



8A=S(X* + rf)ds=^ + d £y xdi/> . (109) 



where a and (3 are given by (14) and (15). The line-integral 

 is to be taken round the entire rim, the double integral over 

 the whole cross section. Employing the double integral we 

 easily deduce 



&A*=^{a-r,)(a* + b*)+fn(l + V )(l*-3z*)}-, . (110) 



whence, if 8 A be the mean value of 8A Z between + I, 



JA/A=(l- v )^p K yE ) .... (Ill) 



where k has the same meaning as in (108). 



Employing the values of a and /3 given by (77) and (78), 

 we obtain precisely the same result as (111) for the change 

 of cross section in the long cylinder. 



Finally, let v = 27rabl denote the volume of the disk, and 8v 

 its increase under rotation, then 



8v=$Adxdydz, (112) 



the integral extending throughout the entire disk. 

 From this we easily deduce by means of (13) 



8v/v={l-2 V )a>*p,cyE, .... (113) 

 or 



8v = co' 2 I/3k } (114) 



where k =E-^{3(1 — 2?;)} is the bulk modulus, and I the 

 moment of inertia of the disk about the axis of rotation. 

 Taking the value of A given by (76), we arrive in the case of 

 the long cylinder at precisely the same formula (114). 



§ 50. Now these coincidences it must be admitted are of a 

 striking character, and if it can be shown that they are not 

 the outcome of mere accident, but the exact results to which 

 the complete theory of rotating elastic solids is bound to lead, 

 this would seem strong evidence of the trustworthiness of the 

 present solution as regards both its soundness of basis and its 



