46 



Mr. J. H. Jeans. 



[Mar. 6, 



2. The hard steels will not withstand a greater number of reversals 

 of the same range of stress than the mild steels if the periodicity 

 of the reversals is great. 



" The Equilibrium of Eotating Liquid Cylinders." By J. H. 

 Jeans, B.A., Isaac Newton Student and Fellow of Trinity 

 College, Cambridge. Communicated by Professor G. H. 

 Darwin, F.E.S. Eeceived March 6— Head March 20, 1902. 



(Abstract.) 



The most serious obstacle to progress in the problem of determining 

 the equilibrium configurations of a rotating liquid lies in the difficulty 

 of determining the potential of a mass of homogeneous matter of which 

 the boundary is given. If this boundary is 



f(x,y,.z) = (i), 



the potential will be a unique-valued function of x, y, and z, of which 

 the form will depend solely upon the form of / (x, y, z). This potential 

 must therefore be deducible by some algebraical transformation of the 

 function /. 



In the method usually followed the solution is found as a volume 

 integral, the integration extending throughout the surface (i). There 

 is, however, a second method of obtaining this potential, namely, by 

 regarding the potential-function as the solution of a differential equa- 

 tion, subject to certain boundary conditions. This leads directly to a 

 series of algebraical processes, enabling us (theoretically) to deduce the 

 potential by transformation of the function /. 



In three-dimensional problems this method is quite impracticable, 

 since it depends upon a continued application of the formula which 

 expresses the 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 present paper deals only, with two-dimensional 

 problems, for which a method is developed enabling us to write down 

 the potential by transformation of the equation of the boundary. The 

 method is not of universal applicability, but is adequate to the problem 

 in hand. 



The method as applied to the determination of equilibrium con- 

 figurations is as follows. Starting from the general equation (in polar 

 co-ordinates) 



r? = «o + 2fl'i?'cos #-f-2a 2 r 2 cos2#+ (ii), 



