of Liquid Surfaces of Revolution. 37 



surface is an unduloid or nodoid according as /3 is positive or 

 negative, i. e. according as 7r/2 > 6 > — 77-/ 2 or 3 7r/2 > 6 > ir/2. 



If be supposed to vary continuously, and if one at least 

 of the quantities a and /3 is finite, the form of the surface may 

 be made to pass through a continuous cycle of changes. 



Thus, between = and 6 — 7r/2 the surface is an unduloid, 

 the limits being the cylinder when = 0, and the sphere when 

 6 — ir/'2. As 6 passes through the next quadrant the surface 

 is a nodoid, the limits being the sphere, and a circle the plane 

 of which is perpendicular to the axis of revolution, which 

 is, as Plateau points out, a purely mathematical limit. In the 

 third quadrant the surface is again a nodoid, the limits of 

 which are the circle and catenoid. Finally, when 6 lies be- 

 tween 37r/2 and 27r the surface is an unduloid, the limits of 

 which are the catenoid and the cylinder. 



If now 2X and 2Y are the distance between and diameter 

 of the rings respectively, and if <£ x is the value of <f> when y = Y, 

 we have 



X = aE + /3F, Y 2 =■■ a 2 cos 2 (/>! + p 2 sin 2 <fr. 



Hence if a, /3, and (f> l vary, but so that X and Y are unaltered, 

 we have, by differentiation, 



f /3 r*> sin 2 A . , _ /3 2 f ?i sin 2 A , , 1 „ a 



+ {«A 1 +£}sfc=0, (3) 



and 



2acos 2 (/> 1 8« + 2 / 8sin 2 </) 1 S/3-( a 2 -/3 2 )sin2^ 1 S^ 1 =:0. . (4) 

 But 



f sin 2 $ , , F - E f sin 2 If f <ty „ ") 



f# « 2 f^ /c 2 sin ^cosfo ) ... 



Substituting these values in (3) and (4) and eliminating Sfa 

 between them we get 



( a 2 E~ / 8 2 F + « 2 A 1 cot</) 1 )Sa + a 2 ( F - E + A i t Wi)^ = ^ 



} ( 6 ) 

 or AS« + BS/3=0. J 



Now the mean curvature of a surface of revolution of mini- 

 mum area has been proved by Lindelof to be the same as 



