ON THE THEORY OF THE CAPILLAEY TUBE
1915]
The integral in (23) can be expressed. We find
o 3a2
C' C ~~ (C" ~"~
355
c2-r2~
6a4
, ~ - ~ c {c + V(c2 - r2)j log
- 1
2c2(log2-l)l......................................(24)
The expression for ra? cos i in terms of c is complicated, and so is the relation between c and i demanded by the boundary condition
coii —
du
.(25)
But in the particular case of greatest interest (i = 0) much simplification ensues. It follows easily from (25) that c = r. When we introduce this condition into (24), we get
(26)
and accordingly
Hence by successive approximations
59""
................ ,..(28)
If the ratio of r to h is at all such as should be employed in experiment, this formula will yield a2, viz., T/gp, with abundant accuracy.
Our equations give for the whole height of the meniscus in the case i = 0, c = r,
-u^r-— (l -^Jlog2
(29)
Another method of calculating the correction for a small tube, originating apparently with Hagen and Desains, is to assume an elliptical form of surface in place of the circular, the minor axis of the ellipse being vertical. In any case this should allow of a closer approximation, and drawings made for Kelvin* by Prof. Perry suggest that the representation is really a good one.
Proc. Roy. Inst. 1886; "Popular Lectures and Addresses," I. p. 40.
23—2ac=r. We must therefore take