845 



will be the relation between the width of the curve and the (sharp) 

 angle at whicii it intersects the .I'-axis '). 



§ 10. Capillary surfaces, whose meridional sections correspond to 

 the curves DFG of fig. 2 or LMN (as also the one of opposite 

 direction) of (ig. 3 can be experimentally realized between two tubes 

 the radii i\ and )\ of which differ comparatively little (say : )\ — 1\ 

 <^ <^ I := }i {)\ 4" ''i))i hut still so much that (r, — r^)l' k is a fairly 

 high number. In general, however, for given values of /> = ^, (r, — 1\) 

 and I = ^ {r^-\-7\), n will not be a whole number, i.e. the surface 

 which establishes itself between two arbitrary lubes does not form 

 part of those discussed in sections 6 and 9. Still, the equations as 

 given ren)ain valid, in other words the raeiidional curve is still 

 represented by equations (14) or (31) and the capillary rise as well 

 as the minimum angle are still given by equations (20') and (35'). 



The 6^-shaped or xS'-shaped curves, obtained in that way, again form 

 part of a series of similar curves, but the first series does not 

 now in general terminate on the side of the axis of revolution in 

 a curve which runs approximately towards the origin, neither does 

 the second series finish up on the side away from the axis in a 

 branch running to infinity. It is easily seen, that the condition is as 

 follows: the (['^-shaped curves become lower and lower on the side 

 of the axis (towards the left) i.e. 2, becomes smaller and after having 

 gone through a minimum z^ becomes imaginary, equation (14) 

 assumes the form (3), that is: the ^/-shaped curves change into 

 ;S-shaped ones. Conversely : the .S-shaped curves assume towards the 

 right smaller and smaller values of ([„^ : ultimately ff',n becomes 

 imaginary and the ^S-curves change into U-curves. ') 



§ 11. We shall now consider the case k<Ci(^ (suspended drops). 



1) Putting 27 = xn — XL, we find 



r/,,= 1,842 e-vt/^. ....... (35') 



2) If (ro — ri)l'A: is a moderate number, the results of the two-dimensional 

 protilem will be applicable as a first approximation without any simplification. If 

 (7*2 ^1^ ^^^ is very small, the section is circular in first approximation and the 

 further approximation may be carried out in a way similar to that used in Suppl. 

 N*'. 42t; for small drops. 



If the difference 72 — ^1 is not small compared to I = ^{ro — t'i^, then in the 

 neighbourhood of .*; = rj, and x = r., the developments of sections 7 and 3 M'ill 

 hold, whereas in the intermediate region 



z^aiB^^'^'ii.vyk) + bJ^{ivi/k), (36) 



the value of a being given by eq. (30) with r — 7\, that of b by // from equation 

 (13) with r = n. 



55 

 Proceedings Royal Acad. Amsterdam. Vol. XXI. 



