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Full text of "Scientific Papers - Vi"

1914]                                     UNDER CAPILLAEY  FORCE                                      263
f
in which p0 should be constant as 6 varies.    In this
cos 6         1 f     r      t       3r2\             r                r2           )
s^_   = i      ^ + (l + fM cos 0--f cos 20 + /-cos 30   , a + r cos 0    a (   2a     \       4a /            2a              4a2          j
l+-cos0   =] + =-- + 0080 +
a        /           2aJ     a             2a2
Thus approximately
r      V/,       r2\           (_      3r2     a?a* 2r
+J + cos 0 a +  - - 
+coe!W.__.    ...(24)
The term in cos 0 will vanish if we take o> so that
T       r The coefficient of cos 20 then becomes
-~+ cubes of-............................(26)
4a                  a
If we are content to neglect r/ct in comparison with unity, the condition of equilibrium is satisfied by the circular form; otherwise there is an inequality of pressure of this order in the term proportional to cos 20. From (25) it is seen that if a and T be given, the necessary angular velocity increases as the radius of the section decreases.
In order to secure a better fulfilment of the pressure equation it is necessary to suppose r variable, and this of course complicates the expressions for the curvatures. For that in the meridianal plane we have
2 _ - ^!r j_ 9 /^rY
1        f         i      r/}2 ~T  /   I   j /. )
JL                   tti/            \CLu/
P~
Ml
or with sufficient approximation
For the curvature in the perpendicular plane we have to substitute PQ', measured along the normal, for PQ, whose expression remains as before (fig. 3). Now
= C08 QPQ> _ tan 0 sin QPQ>
Sin
in which
ON    L     1 /dr\-ione point on the scale a fine pencil point protrudes through a small hole and describes the diminutive circular arc. Another point of the scale at the required distance occupies the centre of the circle and is held temporarily at rest with the aid of a small brass tripod standing on sharp needle points. After each step the celluloid is held firmly to the paper and the tripod is moved to the point of the scale required to give the next value of the curvature. The ordinates of the curve so drawn are given in the second and fifth columns of the annexed table. It will be seen that from x  0 to x = 2 the curve is very flat. Fig. (1).