36 Mr. A. W. Biicker on the Critical Mean Curvature 



than half the circumference of either. If the distance between 

 the rings is exactly half the circumference, an infinitely small 

 change in the volume will modify the form of the surface, but 

 will not alter the mean curvature. Thus the mean curvature 

 of a cylinder, the length of which has this particular ratio 

 (•7r/2) to its diameter, is evidently a maximum or minimum 

 with respect to that of other surfaces of constant mean curva- 

 ture, which pass through the same rings at the same distance 

 apart, and which differ but little from the cylindrical form. 

 Hence the cylinder may be said to have a critical mean cur- 

 vature when the distance between the rings is half their 

 circumference. If the distance between the rings is altered, a 

 similar property w T ill be possessed by some other surface. It 

 is proposed in ^he present paper to determine the general 

 relation between the magnitude and distance of the rings and 

 the form of the surfaces of critical curvature. 



The expression for the change in the mean curvature of a 

 film or liquid mass, under the conditions above laid down, has 

 been investigated in a paper " On the Relation between the 

 Thickness and the Surface-tension of Liquid Films," lately 

 communicated by Prof. Remold and myself to the Royal 

 Society. It was, however, applied only to the cases which 

 were practically realized in the experiments therein described. 

 It will be convenient, before discussing it more fully, to 

 indicate the manner in which the equation is obtained. 



Beer has shown that if the axis of x be the axis of revolu- 

 tion, the equation to a liquid surface of revolution is given by 

 the expressions 



x=x~E+j3F, f =u 2 cos 2 0+£ 2 sin 2 $; . . (1) 



where F and E are elliptic integrals of the first and second 

 kinds respectively, of which the amplitude is 0, and the 

 modulus tc = \/a 2 — {3' 2 lu. 

 As usual, 



A = v / l-^ 2 sin 2 </), (2) 



whence y = uA ; and if k= sin 6, /3 = a cos 6. 



Since a. > /3, a and (3 are the maximum and minimum values 

 of y respectively : and the above equations implicitly assume 

 that the origin lies on a maximum ordinate; for when </> = 0, 

 # = and y — cL. If we wish to transform to a minimum 

 ordinate, <£ is > 7r/2, and 



.r = «(B-E 1 )+iS(F-F 1 ), .... (la) 



where E t and F x are the complete integrals. 



It may be well, for the sake of clearness, to state that the 



