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If the semi-axis minor of the ellipse be &tho curvature at the end of axis is /3/r2, and in our previous notation £ = h?•"/*2«-. Also, / being e< to 0,
and                           a" = i r (h + J£) = AAr (1 + r'im").........
This yields a quadratic in (/.-; hence
,,Jir    hr    /'       4r\     /• f.      r _ r-       :>/•* [
a"~ '4 + "4 V \       MJ     2 V     •*'   S)/'     ~7/<"l
— ^.^ ! //. -|_ i ;• — ()'l HI r"lh -H 0*0741 /^ //'•'; ........
approximately.    It will be seen that this differs but  little
(28), which, however, professes to be the accurate result. s<» far an th'- I-TI
r3//i2 inclusive.
77/6 Wide 'Mr.
The equation of the second order for the surface of the liquid, iiv-iime be of revolution about the axis of z, is well known and may be derive*! 1 (1) by differentiation. It is
If dzfdu; be small, (-S2) becomes approximately
d*z     .1 dz     s      .'te /<l2\*     1 /th r1 die*    x dn:     ii~     "lii1 \dfj      JK \(t.('/
In the interior part of the. .surface under consideration (ds:tLrf mil neglected, arid the approximate solution i«
\         ya           **"*             ^
z- »fl, 0 (ia,/a) = /„ a (•'/") •• *o i    + %2%'J "f" '2s. 4s. «4     ••• j .-••'••
,/o denoting, as usual, the Bessel'M, or rather Ftturier'H, function of /rj-n «« and //(, being the elevation at the axis above the free abwilutfly plane j< For the present purpose /;,, is to be HO small HK to be negligible iti I'spermi and the question is how large must /• be.
When h-Q is small eiioiij/h.a'jti may be large while (h •'*/./• still n-uuiiitH HJ Eventually dzjdx increases so that the formula faik But when .f is f enough before this occurs, we may if inHX'Hsnry carry on with the dimensional solution properly adjusted to fit, JIN will be further rxpla later. In the meantime it will be convenient to give wmie numerical i-xutri of the increase in dzjdx. In the usual notation
and the values of Ilt up to x/a— 6, are tabulated*.
* Brit. Atsoc. Ilep.for 18KU ; or Gray and Mathew»' Itntel't Function*, Table VI.all, the total curvature is nearly constant, that is, the surface is nearly spherical    We take