Solid Cylinders of Elliptic Section. 99 



curvature R of the curve is given by 



PySB={ («„-**>vV + (/3 -/30j//VK V?V+?/6?, 



where the suffixes 0, I denote the corresponding values of z. 

 Having regard to (30), (31), and (32), we easily deduce 

 from (14) and (15), 



R- 1 = 2M(^ 2 /a 1 4 + ( y 2 /V)* (67) 



To put this in a more convenient form, let ka ly k\ be the 

 semi-axes of the ellipse of equal longitudinal displacement 

 through (#, ?/, 0), and let kp 1 be the perpendicular from the 

 centre on the tangent at (%, y, 0) to this ellipse. Then we 

 have 



U~ 1 = 2Mk/p 1 ={-Bl/l)x(2k/ Pl ). . . . (68) 



Thus for points on a given ellipse of equal longitudinal dis- 

 placement R varies as the perpendicular from the centre on 

 the tangent, and for points along a given radius-vector r from 

 the axis of rotation R varies as 1/r. 



§ 32. At the rim it is convenient to measure the curva- 

 tures in lines originally parallel to the axis in two planes, 

 one containing the normal, the other the tangent to the rim. 

 Calling these curvatures 1/R ;i and 1/R^ respectively, and 

 denoting by p the perpendicular from the centre on a tangent 

 to the rim, we easily obtain 



_a fifya 8 2ja- 1 (a i - i 2 V a 2 U 2 + b 4 )+ap- 1 {l + 3 v )b 2 {a 2 + b 2 ) 



R K E ** 3a i + 2a 2 b 2 + 3b 4 ' ^ 



a _co 2 pa 2 9 pxy a 2 — b 2 a 4 — 2ya 2 b 2 + 6 4 ,- m 



Ri~ "E"^ ab 2 ~H 2 ~Ja 4 + 2a2b 2 + Sb 4 (7U) 



From the form of (69) it is obvious that the curvature 1/R W 

 has a minimum where 



p 2 =(l+3y)a 2 b\a 2 + b?)+(a i -2ya 2 b 2 + b i ). 



This is, however, a real minimum only when 



p 2 < a 2 , or a 4 > a 2 b 2 {l + V ) + 3 V b 4 . 



Unless this inequality hold, 1/R 7l is a minimum at the end 

 either of the major or minor axis. The maximum value of 

 1/Rn always occurs at the end of a principal axis, and this is 

 the major or the minor axis according as 



(a-b) 2 (a 2 + ab + b 2 )>ov<r)ab{Sa 2 + 2ab + 3b 2 ). 



§ 33. The curvature 1/R^ may be regarded as measuring 

 the tendency of the material in the central plane at the rim 



H 2 



