Solid Cylinders of Elliptic Section. 157 



(iii.) A normal pressure —aPpr*K! directed along the 

 radius r from the axis of rotation. 



(iv.) A normal pressure directed along the tangent at the 

 point considered to the ellipse which passes through 

 the point and is similar and similarly situated to the 

 cross section. If a, V denote the semi-axes of this 

 ellipse, p' the perpendicular from its centre on the 

 tangent, this pressure is 



2 1 + 27; g%* (g'b'\* ™. 



-»*Pi- v w+2aW+w\T)' ' ' (86) 



On the curved surface of the cylinder, of the two principal 



stresses other than zz one, directed along the normal, is zero, 

 and the other, whose line of action is the tangent to the 

 cross section, is given by the equation 



71- 2~ a* + b*- V (a* + b*f (ah-* 



Z Z — (O " 



?} • ■<«) 



(l~7;)(3a 4 + 2a 2 /> 2 + 36 4 )Vi> 



where p is the perpendicular from the centre on the tangent 

 to the elliptic section. This stress vanishes when the section 

 is circular for the limiting value *5 of rj. Under all other 

 conditions it is a tension. This analysis of the stress is very 

 similar to that given for the thin disk in § 5. When r\ is so 

 small that its square is negligible the two analyses are abso- 

 lutely identical. 



§ 37. The stress zz vanishes over the cylindrical surface 



*7V + </7V =1 , (88) 



where 



a n 2 = l(a 2 + & 2 ){(a 2 + & 2 ) 2 -r^ 



b n * = \(a? + b*) { (a 2 + 1*)* - V (a 2 - 6 2 ) 2 } -J- { 2a 4 + a%* + b* + i?(a 4 -5 4 )}.(90) 



It is a tension at points within, a pressure at points outside 

 this surface. If e n denote the eccentricity of (88) and e 

 that of the rotating cylinder, 



e n ye 2 ={l + 2 V )a^a* + b*) + {2a i + a% 2 + b i + r ] {a 4 --V)}. (91) 



When b = a and rj = '5 then e n — e. But under all other 

 conditions 



e n <e, 



or the surface of no longitudinal stress and the surfaces of 

 equal longitudinal stress, which are similar to it, are of 



