of Liquid Surfaces of Revolution. 39 



where 6 is <7r/2 for the unduloid and >7r/2, but <ir for 

 the nodoid. 



All these forms will have a maximum diameter half way 

 between the rings. If we now proceed to discuss cases 

 where the principal ordinate is a minimum, we must all 

 through the previous investigation consider the lower limit of 

 the integrals to be ir/2 instead of 0, and fa to be > tt/2. With 

 this convention no change is produced in an y of the equations ; 

 as in equations (5) the quantities which are brought outside 

 the sign of integration vanish, both when fa = and when 



fa = 7r/2. 



Thus, writing as usual E x and F x for the complete integrals, 

 and taking fa instead of fa as the upper limit of E and F, 

 where fa is >7r/2, we have 



2(E-E 1 )-(F-F 1 )(1+ cos 2 0) + 2A(<J>') cot 2^ = 0. (9) 

 Let yjr be an angle such that 



F(f)-F 1 = F(f); 

 then, by the addition theorem, 



E(f)-E 1 = E(^)-sin 2 ^sin(/) / sin^. 

 Also 



tan fa tan i|r= — sec 0, 



sin fa = cos yfr/A. (yfr) , 



A(<//)A(»=cos0, 



and 



. n . . 1 — cos 2 6 tan 2 -\lr 



cot 26' = -^ m r- 5 -. 



r 2 cos 6 tan i|r 



Hence, substituting in (9), 



2E(f)-F(f)(l + cos^)-2sm^^g^ 



cos 6 1 — cos 2 # tan 2 yjr _ 

 + A(t) X 2cos0tan^ ~ ; 

 or 



2E(»-F(^)(1 + cos 2 6>) + 2A(.|r) cot2^ = 0, . (10) 



which is the same as (7). 



We thus conclude that, for every angle i/r or fa which 

 satisfies (7), there is a corresponding angle fa which satisfies 

 (9) for the same value of « 2 , and that these angles are con- 

 nected by the relation 



tan fa tan fa — — sec 6 (11) 



If, then, we determine from (7) the values of fa which cor- 

 respond to certain values of between and tt/2, we can by 



