164 Lord Rayleigh on the Equilibrium of 



So long as Q,<l,p is positive and the (equal) curvatures 

 at B are convex. When £1 = 1, p = and the surface at B 

 is flat. In this case (8) gives 



± di}=y/{a*-a*y " • • " ( U ) 

 or if we set x = a sin 3 <£, 



dy a . i , „ _,. 



±4=3 sm * < l5 > 



Here # = a corresponds to <£ = -|7r, and ,v = corresponds to 

 <£ = 0. Hence 



OB=| f 2 sintytty (16) 



The integrals in (16) may be expressed in terms of gamma 

 functions and we get 



OB = avV.r($)-7-r(£)=-4312a. . . (17) 



When 12 > 1, the curvature at B is concave and p is 

 negative, as is quite permissible. 



In order to trace the various curves we may calculate by 

 quadratures from (4) the position of a sufficient number of 

 points. This, as I understand, was the procedure adopted by 

 Beer. An alternative method is to trace the curves by 

 direct use of the radius of curvature at the point arrived aL 

 Starting from (7) we find 



d?y [„Sw 2 1-Q,\dx 

 ds* 

 and thence 



~ \ a* a J ds 



a d 2 y ds 2 3x 2 1 n nQ . 



~=a—/' =12— 0- + 1— 11. . . . (lb) 

 p ax/as or 



From (18) we see at once that 12=0 makes p = a throughout, 

 and that when 12 = 1, ^ = makes p = co . 



In tracing a curve we start from the point A in a known 

 direction and with p = a/(2X2 + l), and at every point arrived 

 at we know with what curvature to proceed. If, as has 

 been assumed, the curve meets the axis, it must do so at right 

 angles, and a solution is then obtained. 



The method is readily applied to the case 12 = 1 with the 

 advantage that we know where the curve should meet the 

 axis of y. From (18) with 12 = 1 and a = 5, 



I- 24 * 2 ' (19) 



Starting from #=5 we draw small portions of the curve 

 corresponding to decrements of x equal to "2, thus arriving 



