of Liquid Surfaces of Revolution. 



41 



The question as to whether the critical value is a maximum 

 or minimum has not yet been discussed. Since A — B = 0, 



this depends on the sign of -^— j -; or, if we write a = A/a 2 , 



da 



b = B/a 2 , upon that of -j- (a--b), where a and b are explicitly 

 functions of and fa only. Now, putting a— b = %, 



d 



dc 



where ( -~ ) and ( -~~ \ are the coefficients of W x and 8 fa, 



in (12), with the signs changed. 



But since A = B, d{3/da=—l } from (6). Hence, since 

 cos =j3/ a j 



la v ' da \dujda \dfaj da 



. Q d0 a+$ 

 da a 1 



In like manner, from (4), 



dfa _2(a cos 2 fa—/3 sin 2 fa) 



so that 



dc 



(a 2 -j3 2 ) sin 2 fa 



^"V^/^sin^" 1 " V#i/ (« 2 -/3 



— /3sin 2 0, 



sin 20! 



Now as we pass from one surface which satisfies the condition 

 A — B = to another, the value of X changes ; and it can easily 

 be shown that if d^K/d0 be calculated subject to this condition, 

 it is of the same sign as dx/da. Hence if X increases with 0, 

 dx/da is positive and the critical value is a minimum ; if X 

 diminishes as increases, it is a maximum. If X is a maxi- 

 mum or minimum the curvature has a stationary value, but 

 it is not itself a maximum or minimum. 



I have calculated by trial the values of fa which satisfy (7) 

 for a few angles between 0° and 90°. They are given, together 

 with the corresponding values of E, F, and A x , in Table I. 



Table I. 



e. 



0i- 



E. 



F. 



A v 



o 







45-00 



* 0-785 



0-785 



1-000 



10 



45-23 



0-787 



0-792 



0992 



30 



4694 



0-799 



0-841 



0-931 



45 



49-28 



0-812 



0-913 



0-844 



60 



52-25 



0-824 



1-018 



0-729 



80 



55-84 



0-833 



1-169 



0-580 



90 



5647 



0-834 



1-200 



0-552 



