79 
OF ROTATING LIQUID CYLINDERS. 
— rt" + (^ + 17 ) + CQ + 1 ?®) +.(46), 
or, in polar co-ordinates, 
r~ = a" -f- 2 «p’ cos 6 + 2a^r’^ cos 2 d -j-.(47). 
This equation is not sufficiently general to represent all cylinders which are 
symmetrical about the initial line. The value of lieing known at the boundary, 
we shall always be able to find a function v such that v is finite and continuous, 
together with its first differential coefficients, at all points inside the boundary, 
and such that =: 0 inside the boundary, and v = at the boundary. The 
value of V near the origin can be expanded in the form 
-|- 2up’ cos 6 + '2a.2r' cos 2d+.(48), 
and this series will have a circle of convergence, say r = R. It is only when the 
curve lies wholly inside this circle that the cylinder can be represented by an 
equation of the form of (47). 
Let us, however, attach a conventional meaning to equation (47) in the case in 
which the right hand becomes divergent at the boundary, as follows, fjet us suppose 
that the value of the function v given near the origin l)y equation (48) is calculated 
from its known values inside the circle r = R, the values' outside this circle being 
obtained by a process of “continuation.” Then we shall suppose equation (47) to 
represent the locus of points at which r" = v. 
Obviously, with this convention, equation (47) is sufficiently general to represent 
any surface. If this surface is to give an equilibrium configuration under a rotation 
(j), we must have 
-L L ^ constant.(49) 
at the surface. Now Yq -|- 7 rp?‘" is a spherical harmonic at all points inside the 
surface, and equation (49) can be written in the form 
( o •. 
1 — ^ j 7 ’^ = a constant, 
or, what is the same thing, 
/ (^2 \ 
(Vj 4 - 7 rp 7 ’") — TT'P (^1 — ) V = a constant.(50). 
This equation is satisfied at the surface S, and each term is a solution of Laplace’s 
equation at every point inside S ; hence the equation must be satisfied at every 
point inside S. 
Now Yi can be calculated by the methods already explained, and we obtain an 
equation of the form 
Y-~C— npY- rrp t f, (fq, a^, .... ) (f'‘+ 77 "), 
?l = l 
which gives the value of Yi at all points inside a certain circle of convergence. The 
value of V inside its circle of convergence is, from equation (48), 
V = <i' ^ a„ (£'* -j- 77 ”) ; 
