138 Mr Basset, On the Application of Lagrange’s  [Nov. 28, 
The integral 
Qr 
=p| |v log ¢ + 2 (2 cos 6 + 25 sin 6)} 
0 
Be | — : (A cos 6 + 3 sin ay} dd = 2apm’ log c. 
Whence = mpm log c+ 4p2 (Ky). . cesses se seen (6): 
Hence we finally obtain 
L=S4+7p> (Av— Bu) +> (No) 
— mpm log c— 4p (Ky) + Vs... (6). 
If we substitute the preceding expression for Z in Lagrange’s 
equations and perform the differentiations, it will be found that 
the terms zpc’m in N, and apm’ log c disappear ; we may therefore 
write 
L=+4+ p> (Av— Bu) + & (No) —49> (ey) + Vi... (7). 
dQ , , 
N=-—ip fr | — mp (Aw + By’).........(8). 
The quantity Z which does not depend on the cyclic motion 
can be obtained by the ordinary methods. With respect to the 
other terms we must first obtain the values of y and ©; we must 
then draw from O a series of lines parallel to the directions of 
U,, U,-»-, and take each of these limes successively as the initial 
line, and expand y in a series of the form 
= — mlogr—*(@ cos 6 + 38 sin 0) +... 
which will determine the values of the A’s and J's. 
The velocities u,v and the co-ordinates 2’, y’ expressed in terms 
of a, y, the co-ordinates of C referred to fixed axes, and the angle 
@ which CA makes with Oz, are given by the equations 
u=acosO+ysind, v=—a#sin@+ycos0 (9) 
zw =axcosO+ysind, y'=—asind+ycos0@) ; 
When there are several cylinders, the value of y at the surfaces 
of the different cylinders is a function of their forms and positions, 
and is therefore a function of the co-ordinates; when there is 
only one cylinder the value of y at its surface is an absolute 
constant. 
We shall now give some examples. 
