358 ON THE THEORY OF THE CAPILLARY Tl'HK
since when -^= 0, & is exceedingly small. According1}'
Also and
The constant is determined by the consideration that ut tin- wall (./•= /•), = tj7r; thus
r - x = log tan (w/8) + V2 - 1< >g tan ( .} -f ) - 2 COM ( A
since •v/r is small.
The value of as is supposed to be the same here as- in (.'*7), HO that
tc = log ^ -f | log (2-Tr.r ) - log /<„ , ..... . .......... . . ( 42 )
whence on elimination of ^ and restoration of (t, r/a = - log (^2 + 1) + \/2 - 2 + 2 log 2 + & log(27r,r:>0 - log(//flVK ...( W)
With sufficient approximation, when A0 in Htnall enough, \vr way here substitute r for x, and thun
r/a -^ log (r/a) = - log (<y/2 + 1) + ^2 - 2 + 2 log 2 4- | log (2w) - !<»g (//»/«) = 0-8381 + log («//*,) ............... . ................ ,..,,..,(44}
This formula should give the relation between r/a and ha/a when hja i« small enough, but it is only roughly applicable to the ease of greatest interest, where «///„ = 1000, corresponding to the accuracy t>f reading f«>imd by Richards and Coombs. In this coao
0-8381 + log («///0)« 7*740.
For this value of r/a, we should have | log (?•/«)« I '024. It. is true that according to (44) r/a will be somewhat greater, but on the other hand the proper value of x (replaced by r) is les.s than r. We may fairly take
rfa = 7-740 + 1-024-8-770,
making with a — 0*27 cm.
2r = 4'74 cm.
This calculation indicates that a diameter greater even than thiwr contemplated by Richards and Coombs may be m'cessary to reilutu- /i,» to negligibility, but it must be admitted that it is too rough to umpire great confidence in the close accuracy of the final number. Probably it would be feasible to continue the approximation, employing an approximate value for the' second curvature in place of neglecting it altogether. But although this integration can be effected, the work is rather long.uation which may be employed when 7i0 is so small that a large oc is consistent with a small ty.