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[Added November 17.—Since this paper was communicated, I have been surprised to find that the problem of the last paragraphs was treated long ago by Laplace in the Mdcanique Celeste* by a similar method, and with a result equivalent to that (44) arrived at above for the relation between the radius of a wide tube and the small elevation at the axis. Laplace uses the definite integral expression for 70, and obtains the approximate form appropriate to large arguments. In view of Laplace's result, I have been tempted bo carry the approximation further, as suggested already.
In the previous notation, the differential equation of the surface may be written
In the first approximation, where the second curvature on the left is omitted, we get
—-—1 _ i _ cos ^ _ 2 sin2 •—, 2a2                  T              2
za being the elevation at the,axis, where -fr = 0.    For the present purpose z£ is to be regarded as exceedingly small, so that we may take at this stage, as
in (39),
z = 2a sin (| ^)...............................(46)
We now introduce an approximate value for the second curvature in (45), writing as = r, where r is the radius of the tube, and making, according to (46),
9       / /           ,j.2  \
sin ^ = -A/(!-/--).........................(47)
T     aV  \      4av
On integration
^2           /L/7   /               >2  \ a           rt<2          A.SI              *f«
^y                     •    __    *             TCt-(/   '                       \                     _L                   8   '                          /'AR^
u-cosy-g^ii+5- I -1 ~Z^J ~ Oo2+ o7cos "o '    ......W
<u\Aj          O/\             TuCt'/             ZJLc/          O/                  «
on substitution in the small term of the approximate value of z.    When •\|r = 0, z* is very small, so that (7 = 1 + 4a/3r, and
— = 2 sin — H__________                                (49)
a             2     3r      sin-^         ..................x    '
is the second approximation to z. From (49) \ dz          ty     a 3 cos2!^. sin2 ^^ — cos
We are now in a position to find x by the relation
* Supplement au Xe Livre, pp. 60—64, 1805..e. a value not much less than r, the radius of the tube. On account of the magnitude of x we have only the one curvature to deal with. For this curvature