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