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



in 



the 



The values in the last four columns are repeated 

 reverse order as 6 increases from 90° to 180°. 



In the next Table are given the values of (f> /, = 7r— <£', and 

 of E 1 ~E(c/> // ) and F 1 — ¥(cf) ,/ ). In representing the results 

 graphically it is best to take cj)'—7r/2 or <j> 2 as corresponding 

 to <p 1} and therefore these values are also given. 



Table II. 



e. 





*>■ 



E.-ECf). 



P 1 -F(f ). 



360 



45-00 



45-00 



0-785 



0-785 



350 



4520 



4480 



0-772 



0-792 



330 



47-20 



42-80 



0664 



0-841 



315 



50-60 



39-40 



0-519 



0913 



300 



57-15 



32-85 



0-326 



1-018 



280 



75-65 



14-35 



0055 



1-169 



270 



9000 



o-oo 



0000 



1-200 



The values of A(^> / ) are omitted because they are readily 

 obtained by the formula A(<£ x ) A(<£ 1 ) = cos 0. 



The curve obtained by means of these Tables, which shows 

 the relation between ^ or <f 2 and 6, is given in fig. I. 



Rectangular coordinates are perhaps the most convenient ; 

 but if 6 and <j) be regarded as angle and radius vector, the 

 curve assumes the symmetrical form shown in fig. II. 



This result completes the solution of the problem ; but the 

 nature of the conclusions at which we have arrived is more 

 evident if we proceed to deduce the ratios of the lengths and 

 principal diameters of the films to the radii of the rings. 



This is done by means of the following relations, where 

 symbols with unity subscript refer to bulging films, and those 

 with 2 subscript to films the principal ordinate of which is a 

 minimum. 



« 1 /Y=1/A 1 , 



X 1 /u 1 = E + ¥cob0; 



whence X x /Y is found. 



/3 2 /Y = a 2 cos 0/Y= cos 0/A 2 = Ai, 



X 2 //9 2 = (E / -E 1 )sec6>+ (F'-F,) ; 



whence X 2 /Y is obtained. 



It is evident from these equations that « 1 /9 2 = Y 5 



e. the 



