130 Mr. A. Ferguson on the Shape of the 



For a liquid o£ zero contact-angle (vii.) gives 



2d 2 r 



^= — — 7v (viii.) 



r 6 



Equation (viii.) gives, to the order considered, the correction 

 for a liquid of zero contact-angle due to the departure from 

 perfect sphericity of the liquid surface inside a capillary 

 tube. In most treatises dealing with the determination of 

 capillary constants one is told that in the case of a liquid of 

 zero contact-angle a correction may be made for the weight 

 of the liquid raised above the plane X'OX (see fig. 1) by 



adding ~ to h, which, as equation (viii.) shows, is perfectly 



correct. But this result is invariably obtained by treating 

 the meniscus as hemispherical, a process which hardly seems 

 logically legitimate, inasmuch as the correction is to be made 

 precisely because the liquid surface is not hemispherical, as 

 is shown by the premisses from which (viii.) is deduced. In 

 fact, the meniscus cannot be treated as hemispherical unless 

 r be negligibly small compared with h (see equation (ii.)), 

 in which case, of course, the correction itself becomes 

 negligible. 



Further, if we calculate the correction by the " weight " 

 method for a liquid of finite contact-angle — that is, by treating 

 the meniscus as a segment of a sphere — we easily obtain 



, 2d 2 . . ,*. . .v r . 2 — sini— sin 2 ^ i 



h= — cos^— r sece 1-sini) \ 1 ^ =-. l , 



r I 3 cos 2 1 J 



which is not at all in agreement with (vii.), but becomes 

 identical with (viii.) when i becomes zero. 



§ 3. It is of interest to obtain for a liquid of zero contact- 

 angle the value of the radius of curvature (R) at the vertex 

 0, and of 77, the value of y corresponding to x=r ; the 

 elementary theory which assumes the meniscus to be hemi- 

 spherical gives, of course, 



r) = r, R = r. 



But from first principles R is given accurately by 



2a 2 



and therefore, substituting the value of h from (viii.), we 

 have 



R 



-(l + £) ..... (ix.) 



