1887.] Hquations to the Motion of a number of Cylinders, &c. 137 
Let (7’, 0’) be polar co-ordinates of a point referred to Oz’ as 
initial line, then at a sufficient distance from O, y can be expanded 
in a series of the form 
we —mlogr’—— @, cos 0 + 98, sin 0’) +... 
Therefore 
2ar 
¥,=- po | {mn log ¢ + : (A, cos 6 + 96, sin 6’) + i, sin 0d’ 
0 
Similarly eee, | | = da’dy’ = — | 7 ds 
S19 0194 (ie RA Sas ABA ROS td SBN N ae (3). 
Again, if ‘N, be the angular momentum about C of the cyclic 
motion, 
( U ne. ! / d. y {/ 
§, =-p[f{(e — aN + y - 9) da'dy 
x 
ot poe ( 7 dx y dy | , / a y 7 
= p | [te a Y dy’ da dy 7p (As, a GY, ). 
By Stokes’ theorem the double integral 
dy dy 
eat 2 YX ik ly) ¢ 
-- ip 7 as + tp fr 7 ds 
The first integral = mpc’m, the second integral may be written 
a | [ranjas.as | 
hence 
2 » dO, / 7 
N,=pe'm — 4p |r as es — mp (A, + ,y,)----.--(4). 
ala x dy 
Also 28 = p |x Sas—p| |x ie as | 
AEDT? dQ, 
= po x Pao + | [Ge as | 
Qa 
= pe| x of dO + p= (xy). 
0 i 
