Solid Cylinders of Elliptic Section. 161 



has a constant value along the radius. But for all other 

 possible combinations in the values of b/a and v) the radial 

 strain along the major axis continually diminishes algebraically 

 as the distance from the centre increases. At the centre the 

 radial strain along the major axis is always positive, but 

 under certain conditions it may be negative, i. e. a com- 

 pression throughout a small length at the ends of the axis. 

 These conditions are most easily investigated by determining 



dot 

 the points where -7- vanishes in the major axis. From the 



symmetry we need only consider the point on the positive side 

 of the axis of y, and we shall denote its abscissa by x . When 

 x Q > a tbe radial strain is an extension along the whole semi- 

 axis; but when x Q <a, this strain is a compression throughout 

 a length a — x at the end of the axis. The expression for 

 x /a given by (77) may be thrown into the form 



(^-a^ + v){^ + ^ 2 + b 4 -r ) (2a 4 + a% 2 + 3b^+ V %a i -b^} 

 = T2 V [(« 2 - & 2 ) (3a 4 - 4a 2 6 2 + 9V - S v * (3a 4 4- 4a 2 6 2 + V)} 



+ (399-l)(3a 6 + 5a 4 & 2 +13a 2 Z> 4 + 36 6 )]. . . (92) 



"Remembering that tj cannot exceed "5, we see that the co- 

 efficient of «r 2 — a 2 in (92) is essentially positive. 



When tj = we have obviously x = a for all values of b/a, 

 i. e. the radial strain vanishes at the end of the major semi- 

 axis, and at every other point of it is an extension. 



When 7] is very small, and b/a is not very small, we find 

 from (92), neglecting terms in rj 2 , 



'"^ f+rfy+y (93) 



The radial strain is a compression throughout this very small 

 length a — x Q of the major semi-axis and elsewhere is an 

 extension. When rj and a are constant, a— ^ has a maximum 

 value rja/3 when b/a = l. 



When b/a is very small as well as 77, we find that the radial 

 strain is an extension over the whole major axis when 



b/a < ^ \^3rj approximately. 



When 77, though no longer very small, remains less than *3, 

 the radial strain is an extension throughout the whole major 

 semi-axis when b/a is small, but is a compression over a small 

 portion a-—x at the end when b/a exceeds a certain value 



Phil Mag. S. 5. Vol. 34. No. 207. Aug. 1892. M 



