140 Mr Basset, On the Application of Lagrange’s [Nov. 28, 
Therefore n = log 2r, — ~ (x cos 0, + y' sin @,), 
whence QA=—x«a' 20, W=—Ky'/2r. 
2 2 2 dQ, 
Therefore N=txp(a +) 4p |r ds 
Let & and © be polar co-ordinates of a point on the boundary 
referred to Cas origin, and CA as initial line, and let OCA =e, 
rP=ae+tyt R—2R Jax’ +’ cos (e+ O) 
= 2+ y’ + a? (cosh’7 cos’ + sinh’7 sin’ &) 
— 2a ./a? + y" (cos € cosh n cos £ — sin e sinh 7 sin £), 
: @) Qa 
whence 4 I 1 ae dss Ss i rd&=k xp (x+y) +3 Kpa’. 
0 
Hence N = — $xpa’, a constant. 
The modified function may therefore be written 
L=Z + hep (y’'u— a'r) =} (Put + Qu? + Cw) + hp (dy — ga), 
where P and Q are the effective inertias parallel to the major and 
minor axes of the cross section, and C is the effective moment of 
inertia about the axis of the cylinder. 
The equations of motion are therefore 
a (Pu cos @— Qusin 0+ xpy) = X, ; 
at | 
£ (Pu sin 0+ Qvcos@—xpx)=YV, ¢ oo... ....c.0s- (GD, 
dw 
Cy (P-@) w=, } 
where X, Y are the components of the impressed forces parallel to 
the fixed axes, and N is the impressed couple about the axis of 
the cylinder. 
Two Cylinders. 
5. Let us now consider the motion of two equal cylinders 
round which there is circulation in opposite directions, and which 
are initially projected with equal velocities parallel to Ox. 
Let A and B be the common inverse points of the two cylinders, 
a the radius of either of them, uw, v and wu, —v their velocities 
