364 



between the planes passing through point of inflection and neck is 

 found to contribute a negligible amount to the total. 



In connection with this it follows from eq. (2') in fourth 

 approximation : 



u,.zi = l/|lR/(l-^it!„'), en w-',= ±kH,' {I -^ kR,') . , (19') 

 the upper sign corresponding to the u[)per index. 



§ 10. Starting from the points D and D' (fig. 3) the analysis may 

 be further continued in a manner similar to the one used above. 

 Indeed the meridian curve of the complete capillary surface consists 

 approximately of a series of nearly semi-circular arcs connected 

 each time by parts of an onduloid or nodoid ^). The centres of these 

 arcs are situated at the heights /?„, 3 R^, 5 /?,, etc. successively ; 

 with each (?i''') arc we therefore place the origin in the corresponding 

 (?i''') centre and as in ^ 2 wi-ite : 



.V = Q sin >'/ , y = (2« — I) li, -o cos >') , Q = R^ C^—^) • (26) 



T is determined by : 



t" sin !) + t' cos >') + 2t sin >') = kh\* {2n—l—cos ,'J) sin Ï) , (26') 

 whence it follows, introducing the condition that the arcs and interme- 

 diate pieces form a continuous curve: 



T--[i i i(n-l)-i^ + i%2 4-in(n--l)- 2:^^i^o^(n-l)- 

 — \n (n - 1) log (± \ kR^^)\ cos /> + - cos Ö log {\ + cos ■'') — 



—^hos>'JloQ{l-cos<'J)]kR,* (27) 



o 



For the connecting curves equations (17), (22) and .r, = xc= — • 



are each time satisfied. 



The successive arcs and their connecting curves can only be 

 realised in separate parts, for instance between two horizontal plates 

 or between two vertical coaxial cylinders. Not every surface, however, 

 obtained in that way is a part of the surface whose meridian-section 

 was analysed above by approximation. As an instance, if the surface 

 is formed between two cylinders which are moistened by the liquid, 



the fraction represents the ratio between the radii of the 



cylinders and this fraction cannot in the analysis of ^ 10 assume 

 any arbitrary (small) value, as long as n represents a whole number. 

 Still, putting 2 {n — 1) ^= a and admitting an arbitrary (positive or 



1) Cf. WiNKELMANN, loC cit., p. 1141, fig. 404. 



