38 Mr. A. W. Riicker on the Critical Mean Curvature 

 that of a circle of radius (a +/3). Hence 



A-B 8a A-B 8/3 



2 8 (r + w) = 



B (a + /3) 2 - A (« + /3) 2 * 



Hence the mean curvature has in general a critical value 

 when A— B = 0. 



First confining ourselves to the case in which the prin- 

 cipal ordinate is a maximum, and <j> l and 6 are less than 

 7r/2, it is evident that, since F is always >E, B is always 

 positive 



Also, by (5), 



whence, since dF/d/c is positive and -sin 2 cos fa is positive, 

 a 2 E — /3 2 F is positive, and therefore A is positive also. Further, 

 B can only vanish if fa = ; and none of the terms in A or B 

 become infinite unless fa = or tt/2, cases which it will be 

 seen hereafter it is unnecessary to consider. 

 Thus, 



(A-B)/a 2 = 2E-F(l+ cos 2 0) + 2A 1 cot 2<£ 1 = . (7) 



is a relation which must be satisfied by (^ and 6 when the 

 mean curvature has a critical value for changes in the form 

 of the surface which take place, subject to the conditions that 

 the radii of the rings and the distance between them are 

 constant. 



Corresponding values of fa and 6 must be found by trial ; 

 but it will now be shown that if such a pair of values is known, 

 when 7r/2 > #>0, the values of fa which are proper to ir— 6, 

 7T + 6, and 27r — 6 can be readily deduced without further trials. 



In the first place it is evident that, since the squares of the 

 sine and cosine of 6 alone enter into (7), the curve obtained 

 by plotting the values of 6 as abscissae and those of fa as 

 ordinates is symmetrical with respect to the ordinate = 7r/2, 

 and that the same value of fa corresponds both to 6 and ir—6. 



If, then, we conceive a film attached to two rings, the 

 volume and length of which vary continuously in such a way 

 that (7) is always satisfied, as the cylinder changes to the 

 sphere and thus to a nodoid forms which correspond to the 

 same value of sin 2 6 will have the same value of fa also, and 

 the lengths will be given by the expression 



X = a(E+ costfF) ; (8) 



