352 ON THE THEORY OF THE CAPILLARY TUBE [399
Also, if h be the height at the lowest point of the meniscus, the quantity directly measured in experiment,
h=l-c ..................................... (5)
In this approximation r/c = cos i, and thus in terms of c
When the angle of contact (i) is zero, c = r, and
the well-known formula.
When we include u, it becomes a question whether we should retain the value of o, i.e. r sec i, appropriate when the surface is supposed to be exactly spherical. It appears, however, to be desirable, if not necessary, to leave the precise value of c open. Substituting the value of z from (3) in (1), we get, with neglect of (dujdos}\
C *~" Ob I / IAJM.' \ C I vJU \ C *""" *JG" J """ C 1 . I .
-**-(&) =^LT+ 3+lwdx\ -(8)
ec xc- c \e/ asc _
For the purposes of the next approximation we may omit (dujdxf and the integral, which is to be divided by a?. Thus
dx V2a2 ' (& ~ a?)* :W so (c2 -and on integration
, °3 /m
T -- , ......... (y>
We suppose with Poissori and Mathieu that
_c^__c^_
c3 so that u ~2 log {c 4- \/(c2 - Ğa)} + (
.. du c3 \/(c2 of) c
corresponding to -7- = -r-,,------77-7-------<-
fj 'jf1 *j£f or \/ (o -~ flu )
To determine c we have the boundary condition
* cot i = ( -J-
r fdu\
dcc/x^r V(c2 T2) ' \dccjx=r
______r ( c3 c V(c2 -
~ A/(c2 - r2) ( ~ .So2 r9" ~~
which gives c in terms of i and r. Explicitly
r rs sin2 i
c
cos i 3 a2 (1 4- sin i) cos3 i These latter equations are given by Mathieu.ibrium of the cylinder of liquid of radius as. At the wall,, where x = r, ty assumes a given value ($TT~ i), and (1) becomes