124 Mr A. B. Basset, On the Application of Lagrange’s [Oct. 31, 
therefore 
do 4 = (, 40 
> (Xu) = 8 (w Tu ee +) + 6, ( a ) + fe. 
= > (06) 
whence (17) may be written 
= © a el) SESE) oP. aoe eee (18). 
4. We have therefore obtained a form of Lagrange’s equations, 
which can be employed when the kinetic energy is expressed in 
terms of the velocities corresponding to the coordinates by which 
the position of the system is determined, and the constant momenta 
corresponding to the time fluxes of the ignored coordinates. Now 
in the hydrodynamical problem we are considering, the product of 
the circulations and density of the liquid corresponds to the 
constant momenta «, «,..... Hence in order to determine the 
motion of a number of perforated solids im an infinite liquid, we 
must first calculate by means of (1) the quantities J and ®; the 
former of which is the kinetic energy due to the motion of the 
solids alone, and is therefore a homogeneous quadratic function of 
their velocities, and must be expressed in terms of the generalised 
coordinates and velocities of each solid; and the latter of which is 
a similar function of the circulations. The quantity X¥ in (18) is 
evidently the generalised component corresponding to u, of the 
momentum of the cyclic motion which remains after all the solids 
have been reduced to rest, and its value is given by (4) or (6), 
according as it is in the nature of a force or a couple. 
5. We can now ascertain the physical meaning of the gene- 
ralised velocity + which corresponds to the momentum xp. 
Let wy, be the flux through the aperture o, of 8, relative to 
S, Then ifl, m,, n, be the direction cosines of the aanine | to o, 
w=] | it - 1, (u, + 2 — ry) —m, (v, + 7,£ — p,Z) 
=, (Ww, +p a.)| do, 
=| dco +{ ae do,— (u,&, at Ot wo, + p,A,+9,4,+ 1V,)/K,P- 
But —— do, = (k,K,) &, + («,«,') «,) + 
