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Full text of "Scientific Papers - Vi"

264                            THE EQUILIBE1UM  OF EEVOLVING LIQUID
approximately,
[387
1   /o5r\2
Thus
COS0
PQ'    a + r cos 0
Idr
sin.0     1 <ir j___1_
J9) \ ^ a~+r co0 r d0 1       2^
.(28)
Fig. 3.
It will be found that it is unnecessary to retain (drjd0)", and thus the pressure equation becomes
2 _      T r ~ r         r
acos6>          asinfl    1 dr    2aa /      rcos0y
+ r cos 0 + a + r cos 0 r 'd0 ~ ~W \   "^      a~ / '
.........(29)
It is proposed to satisfy this equation so far as terms of the order r2/a2 inclusive.
As a function of 0, r may be taken to be
r = r0 + Sr = r0 + rx cos 0 + rz cos
.(30)
where r1} ra, &c. are constants small relatively to r0. It will appear that to our order of approximation (8r/r0)2 may be neglected and that it is unnecessary to include the r's beyond rs inclusive. We have
It/        Lt/    ] T           w'           /        \   I
a cos 0
= _        _     . + cos
a + r cos 0        2a
/       r cos 0\8 _ 1     r1!
2a
4. PA-Q     2r a. r2                 .
r COS O "\ - ~r 
a      cfc      2a2      2a2
+ cos 20 \-l + - + 5^- + ~^-H + cos30 J-" + -2- + J
a     2a2     a2 j                (a      a2      2a2] to suppose r variable, and this of course complicates the expressions for the curvatures. For that in the meridianal plane we have