1887.] Elastic Solid in Polar and Cylindrical Coordinates. 117 
external radius however small be the radius of the inner surface. 
The second solution embodies a general treatment of the equa- 
tions, involving an extensive application of Bessel’s and analogous 
functions. It is found usually to necessitate certain relations 
between the forces on the flat ends and on the curved surface, 
and its application to cases where the surface forces are arbi- 
trarily assigned presents difficulties which have not been sur- 
mounted. It solves however certain problems of interest, the 
most important being the torsion of a cylinder when the forces 
causing torsion are any given function of the distance from the 
axis. The case when the force varies as r* is worked out as a 
special example. 
(4) A Table of the values of e* for values of x between 0 
and 2 increasing by ‘001. By F. W. Newman, communicated by 
Prof. ApAms, M.A. 
Prof. Adams gave an account of the methods employed in 
constructing and verifying the Table which is being printed in 
full in the Transactions of the Society. 
(5) On the Application of Lagrange’s Equations to the Motion 
of Perforated Solids in a Liquid when there is Circulation. By 
A. B. Basset, M.A. 
1. When a number of perforated solids are moving in an 
infinite liquid, and there is circulation through the apertures of 
the solids, the kinetic energy of the solids and liquid (as will 
be proved later on) is equal to the sum of two homogeneous 
quadratic functions of the velocities and circulations respectively. 
Now when a liquid of density p occupies a multiply connected 
region, circulation « can be generated by the application of a 
uniform impulsive pressure xp, applied to any one of the barriers 
which must be drawn to render the region simply connected ; 
and if once generated, it cannot be destroyed excepting by the 
same process as that by which it has been produced. It therefore 
appears that the quantity xp is a quantity in the nature of a 
generalised component of momentum. 
Now the kinetic energy of a dynamical system is expressible 
in three forms, (1) a homogeneous quadratic function of the 
generalised velocities, which is the Lagrangian form; (11) a similar 
function of the momenta, which is the Hamiltonian form ; (i11) a 
mixed function of the momenta and velocities, which is called 
by Routh the modified form. The Lagrangian form is the only 
one which can be used in forming Lagrange’ S equations; and since 
