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 co¥0 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