118 Mr A. B. Basset, On the Application of Lagrange’s | Oct. 31, 
the kinetic energy in the present case is expressed in the mixed 
form (i11) it will be necessary to obtain a modified form of the 
Lagrangian function, which enables Lagrange’s equations to be 
written down without a knowledge of the generalised velocity 
corresponding to the momentum xp. The principal object of the 
paper is to determine this function in such a form, that it may 
be calculated whenever the velocity potential of the liquid is 
known; and the result is expressed in terms of the circulations, 
the velocities of the solids, and the surface integrals of the 
constituents of the velocity potential taken over the boundaries 
and barriers of the liquid. Finally it is shewn that the generalised 
velocity corresponding to «p, where « is the circulation round any 
closed circuit which cuts any particular aperture of one of the 
solids once only, is equal to the flux through that aperture relative 
to the particular solid*. 
2. The following notation will be employed. 
@ = velocity potential of the whole motion. 
W= do. due to motion of solids alone. 
Q= do. due to cyclic motion. 
Uns mr Wms Pin> Im> Tm, the linear and angular velocities re- 
spectively of any solid S,, along and about axes fied in the solid. 
Pins Pm > Pm > Xm>Xm > Xm > the velocity potentials of the liquid, 
when the solid S,, is moving with linear and angular velocities 
respectively along and about axes fixed in S,,, and all the other 
solids are at rest and there is no circulation. 
/ ” 
Om) Om > Tm ++» the areas of the apertures of S_,. 
ay ae tate the circulations through them. 
On, ®,,,@,, --- the velocity potentials due to unit circulations 
through the ere of S,, when all the solids are at rest. 
m? 
Wr Von’ y,,”... the fluxes through the apertures of S,, 
relate to WN,,. 
®@,, the velocity potential due to the motion of S,, and the 
circulations through its apertures, when all the other solids are 
at rest; so that 
®,, aes UmP m ab UmP ak WP mn + DinX m ale Quine te (eg Je Kn 
cn Ona 
* This theorem was discovered by Sir W. Thomson. Proc. R. S. #. Vol. vu. 
p. 668. 
