166 Lord Bayleigh on the Equilibrium of 



When fl>2'4 and the curve does not meet the axis at all, 

 the constant in (3) must be retained, and the difficulty is 

 much increased. If we suppose that dyjds — + 1 when 

 x = a 2 and dyjds =—1 when x = a lt we can determine p as 

 well as the constant of integration, and (3) becomes 



dy 



(TO)' 



x ts- 8T { * 2 ' 



■ ai 2 X^-a 2 *) + 



■a x a 2 



(20) 



a 2 —a i 



We may imagine a curve to be traced by means of this 

 equation. We start from the point A where y = 0, x = a 2 and 

 in the direction perpendicular to OA, and (as before) we 

 are told in what direction to proceed at any point reached. 

 When «£ = <*!, the tangent must again be parallel to the axis, 

 but there is nothing to ensure that this occurs when y = 0. 

 To secure this end and so obtain an annular form of equi- 

 librium, <tco 2 /T must be chosen suitably, but there is no means 

 apparent of doing this beforehand. The process of curve 

 tracing can only be tentative. 



If we form the expression for the curvature as before, we 

 obtain 



8TV 



3a? 2 - 



a i —a 2 



■ «i 2 a 2 2 \ 

 a* ) 



a } a 2 



(21) 



a 2 — cti x 2 (a 2 — a{) 



by means of which the curves may be traced tentatively. 



If we retain the normal PQ, as we may conveniently do 

 in using Boys' method, we have the simpler expression 



^ + -2-. . . (22) 



1 1 <™ 2 /0 2 2 



4T 



a 2 — ai 



When the radius CP of the section is very small in 



v 



comparison with the radius of the ring OC, the conditions are 

 approximately satisfied by a circular form. We write CP = r y 



