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II. On the Equilibrium of Eotatinf/ lAquid Cylinders. 
By J. H. Jeans, B.A., Isaac Newton Student and Felloiv of Trinity College, 
Ca rn bridge. 
Coinmunicated by Professor G. H. Darwin, F.R.S. 
Received March 6,—Read March 20, 1902. 
Introduction. 
§ 1. As a preliminary to attacking the problem oi determining the equilibrium 
configurations of a rotating mass of liquid, I was led to consider whether some 
method could not be devised for calculating the potential of a homogeneous mass 
in a manner more simple than that usually adopted. What was obviously required 
was a calculus enabling us to write down the potential of such a mass by an 
algebraical transformation of the equation of its boundary, instead of by an 
integration extending throughout its volume. 
There was found to be no difficulty in reducing the calculation to a problem of 
algebraical transformation, but in tliree-dimensional problems the transformations 
required were, in general, as impracticable as the integrations which they were 
intended to replace. This was because the transformations depended upon a 
continued application of the formula which expresses tlie products or powers of 
spherical harmonics as the sum of a series of harmonics. 
As soon, however, as we pass to the consideration of two-dimensional problems, 
the spherical harmonics may be replaced by circular functions of a single variable. 
The transformation now becomes manageable, and for this reason the })resent pa])er 
deals only with two-dimensional problems. 
The first part of the paper contains a short sketch of a theory of two-dimensional 
potentials. I have, however, confined myself strictly to such problems as are 
required for the solution of the main problem under discussion, namely, that of the 
rotating liquid; the method does not attempt to be one of general applicability. 
The Potentials of Homogeneous Cylinders. 
General Theory. 
§ 2. We shall suppose the cross-section of the cylinder of which the potential is 
required, to be bounded by a single continuous curve S enclosing the origin. Let 
(322.) K 2 2.5.11.02 
