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Full text of "Scientific Papers - Vi"

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