120 Mr A. B. Basset, On the Application of Lagrange’s [Oct. 31, 
The above equations at once follow from Thomson’s extension 
of Green’s Theorem. For if in equations (30) and (81) of Lamb’s 
Motion of Fluids, Art. 66, we put d= 0, ~=¢,;, then since a, is a 
monocyclic function whose increment is unity for all circuits which 
cut the barrier o, once, and zero for all other circuits, and @, 
is a single valued fanction, we obtain 
AM MEM Te Gey 
[[e. S as || do, ={] $/7* as, 
Now d¢,’/dn is zero at the surfaces of all the solids except S,, 
and da,/dn is zero at the surfaces of all the solids (see Lamb, Art. 
120), whence the third of equations (1) follows at once. The others 
can be proved in a similar manner; it therefore follows that 
(Pe SS See 
where & is a homogeneous quadratic function of the velocities of 
the solids alone, and § is a similar function of the circulations. 
If p be the pressure and /,, m,, n, the direction cosines of the 
normal to S,, 
X,= Mu, + | | plas, 
d Uy 
=Mu,-p |] ¢ = ds, . 
But ne = (u,u,) U, + (Uv,) ¥,+ 
1 
=-p|[(uditug’+ ) oP: as, 
=p {fo as, +p |[o "Pas, 
=~p [fp as, +p [[ 5 (edo), 
where the summation refers to corresponding products, and ex- 
tends to all the barriers; hence 
Y dbs D 
ae tn | [¢ SHOT... (2). 
Therefore 
Le =F 4 b= Fp [| Gt § de) + Bn @) 
oF ae hag eee : 
where — = &=p | x, («ldc) , 
and the summation >, extends to the barriers of S, only. 
