68 MR. J. H. JEANS ON THE EQUILIBRIUM 
V be the potential of thi.s cylinder, supposed to be composed of homogeneous 
matter of density p. 
The value of Y must be finite and continuous at all points except infinity, and 
its first differential coefficients must also be finite and continuous at all points. 
Also Y must satisfy V“ = 0 at all points outside S, and V- Y = — 47rp at all 
points inside S. At infinity Y must vanish, except for a term proportional to log r. 
These conditions suffice to determine A" uniquely. For if there were two distinct 
solutions A" and A^', the function (A^ — A"') would satisfy [Y — A"'} = 0 at all 
points of space, would be finite and continuous, together with its first differential 
coefficients, at all points of space, and would vanish at infinity, except for a term 
proportional to log r. The only solution satisfying these conditions is known to be 
A^ — Y' = 0, hence any function A" satisfying the conditions laid down above must 
be the potential of which we are in search. 
§ 3 . Let us use polar co-ordinates r, 6 in conjunction with orthogonal co-ordinates 
X, y, and let us also introduce complex variables f, rj, defined by 
^ = re‘^ = X + iy, rj — re~‘^ — x —iy. 
Let us suppose the equation to the curve S to be written in the form 
.( 1 ). 
and let us imagine that this equation is solved explicitly f a’ ^ in the form 
^=F(''7).(:^)- 
Tn general F (17) will be a multiple-valued functic)n of y, and the equation (l) 
may, and probably will, be satisfied for other values of f and p than those which 
occur on the surface S. 
Let us, however, suppose that we have succeeded in finding one value of F (77) 
such that this value is a single-valued function of y at every point of S, and is 
equal to Let us suppose that we have succeeded in expanding this value of F (77) 
in a series of ascending and descending powers of 77, these series each being supposed 
convergent at every point of S. Let us wiite 
F ( 77 ) = (^( 77 )-f 1 / 7 ( 77 ).(3), 
(f) (77) == Oq + a^y 4 - a,y^ + . . . ( 4 ), Q (77) = ^ -f -F . . . . ( 5 ). 
AA^e shall consider only the case in vvlheh the surface S has the plane d = 0 as a 
plane of symmetry. In this case the equation / y) — 0 remains the equation to 
the curv’e after the sign of 6 is changed, and we therefore hav^e as a second form 
of this ecpiation, 
= .(6). 
There is therefore a solution of this equation expi'essing 77 explicitly as a function 
of ^ in the form 
y = F (i) = (f) {()-r .( 7 ) 
where F, cj), xp are defined by ccpiatlons 3 , 4 and 5 . 
