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1915]         ON THE THEOEY OF THE CAPILLARY TUBE                        353
We have now to find the value of a2 to the corresponding approximation. For the observed height of the meniscus
',    ............ (16)
and
aV cos i - \  zxdx = ~ (I + 0) + ^ {(c2 ~ r2)1 - c8} + f (u - C) xdx
JO                 &                    "                                 Jo
In the important case where i  0, the liquid wetting the walls of the tube, c = r simply, and
_ r L     r     2?-2 A     0     1 2i (_       3     3/t \            2
= i?;(;i + |r-0-1288 r2//0......................(18)
This is the formula given long since by Poisson*, the only difference being that his a" is the double of the quantity here so denoted.
It is remarkable that Mathieu rejects the above equations as applicable to the case i = Q, c = r, on the ground that then dufdx in (13) becomes infinite when . = ?". But d*/(rz  ^/(i^with which dujdx comes into comparison, is infinite at the same time; and, in fact, both
in equation (8) vanish when x = r.     It is this circumstance which really determines the choice of I in (11).
We may now proceed to a yet closer approximation, introducing approximate values of the terms previously neglected altogether.    From (13)
fjll\2          CB                                                                             3.
) =-22---2-2-2--j
and from (12)
* uxdx = \Cx* + - [tf log {c + V(c2 - *2)} + ^c2 - c V(c2 - 2) + i (c2 - a?)].
0
* Nouvelle Th6orie de V Action Capillaire, 1831, p. 112. B. vi.                                                                                                                23ccjx=r