365 



negative) value for k, the equations (26) and (26') remain valid and 



T = kR^' [a I ^ (t -|- b coi^ d -f -i- (-| — a) cos log (l -f- cos I'J) — 



— \ {\ — a) cos ,') log {\ — cos d)\, . .... (28) 



where a and /; are integration-eoiistanls. R^ is still undetermined, as 



also h, which remains connected with R^ through the relation 



2 

 M = — ; as regards the value of a, this may be chosen at wilP). 



With small values of >'J the curve shows a minimum for y or a 

 point of inflexion') according as {^—a)k^O; for a value of 

 which differs but little from n the curve has a maximum for y, if 

 {\ — a) k'^ or a point of inflexion, if (f — a) k <^ 0. ') 



^12. Here again the meridian-section consists of a series of curves 

 which, however, now extends indefinite!}' upwards as well as down- 

 wards. For k'^ the higher curves in the 

 series show maxima and minima for y, the lower 

 ones points of inflexion, as represented diagram- 

 maticallv in fig. -4. For /; ^ on the other 

 hand the upper curves have points of inflexion 

 and the lower ones maxima and minima of y, 

 which case is obtained by turning fig. 4 upside 

 down. Putting \ — a = ^ the successive minima 

 and maxima of a; satisfy the relations 



z=izh 



-= R. 



2n\ 

 /3 + -j^i2/ 



1 2n+l' 

 - a—^ H ~-~ 



2 6 



2n 



kR. 



At the point where ^3 -\ changes its 



(29) 



sign 



Fig. 4. 



(smallest value of ./!„„„) is the transition between 



2m 

 the two kinds of cui-ves. If accidentally ,d = -^, ???, being a whole 



number, the smallest value of .i'„„„ becomes zero and the case reduces 

 to tliat of the meridian-sections discussed in § 10. 



1) Supposing for instance the meniscus to be formed between two co-axial 

 cylinders which are moistened by the liquid, the radii of tlie cylinders being E 

 and r, where r has to be small with respect to R a and R^ are determined by 

 the conditions xr = r and Xa = R', y- and h may still be chosen at will; one 

 might for instance take a = 0, while determining h by putting yu = 0. 



2) In general therefore in this case the presence of a minimum or maximum 

 for y is not, as in the section 6 sqq, bound to /c > or llie existence of a point 

 of inflexion to k <0. 



24 



Proceedings Royal Acad. Amsterdam. Vol. XXI. 



