142 Mr Basset, On the Application of Lagrange’s _[ Nov. 28, 
The function R has been determined by Mr W. M. Hicks*, and 
also by Prof. Greenhill+, and its value is 
Bm mip [L420 — 98, my 
where g =e **. 
If we suppose the cylinder B to be replaced by the fixed plane 
Ox which forms the boundary of the liquid, the value of L must 
be halved, and the equations of motion of the cylinder A will be 
aes 
* du 
dt 
ddd.) as de, « da 
+ xpe) 2X. \e te (13), 
fads? Gy ee da Mea AOC OOO O00 (14). 
Now c=Jy’—a’ and y=acosha, 
Therefore de = cotha, da = 1 ; 
dy dy c 
whence (14) becomes 
d d& dd 2 m 
ai do by Pu cothat = nee (15). 
Let us now suppose that gravity is the only force in action, and 
that the plane boundary Oz is horizontal, forming, so to speak, the 
bed of the ocean; (13) and (15) respectively become 
Ru + kpc = const. = h 
, f dR K ‘ -»i LOE 
kRi+k(v [pice ee ae 
These equations are satisfied by v=0, u and y constant, pro- 
vided w satisfies the quadratic 
pu? —spucoth a+ © +(M—M’)g=0.0.....-, (16), 
where p=—4dR/dy. The roots of this quadratic will be real 
2 
provided Kp’ coth? a > p a +4(M—-WM’) o| ashe aes Cig): 
* «On the Motion of Two Cylinders in a Fluid,” Quart. Journ. Vol. xv1. 
+ ‘Functional Images in Cartesians,” Ibid. Vol. xvii. pp. 356—362. 
