TRANSACTIONS OF THE SECTIONS. 23 
The object of the present paper is to examine whether we can assert such an 
equivalence to hold approximately, and if so, to what degree of approximation. 
The question was started some years ago by the Astronomer Royal in the ‘ Astro- 
nomical Notices.’ 
In the course of a discussion of M. Delaunay’s views, he proceeds to remark :— 
“Tt will probably be difficult to say what is the effect of friction in more compli- 
cated cases. Conceive, for instance (as a specimen of a large class), a tide-mill 
for grinding corn. The water which has been allowed to rise with the rising tide 
is not allowed to fall with the falling tide; but after a time is allowed to fall, 
thereby doing work and producing heat in the meul formed by grinding the corn. 
I do not doubt that this heat is the representative of vis viva lost somewhere ; 
but whether it is lost in the rotation of the earth or in the revolution of the moon, 
T am quite unable to say.” 
Tt oceurred to me that considerable light might be thrown upon the whole 
subject by combining the equation of energy with that of the conservation. of 
angular momentum. 
Let us first suppose the case of a binary system, consisting of the earth and moon, 
the orbit of the latter being supposed to coincide with the earth’s equator. 
Let Q denote the energy which during a given interval passes into heat through 
tidal action; then, assuming the moon spherical, and her rotation consequently 
unaltered, we have ; 
Q=-—A (energy of-earth’s rotation) —A (energy of lunar orbit). 
By the energy of the lunar orbit is meant the kinetic energy of the revolution of 
the earth and moon round their common centre of gravity, together with the 
potential energy of their separation. 
Now such energy of orbit = const—Zmm'p . is 
a 
where 72, m’ represent the masses of the two bodies, 
p the attractive force at unit. of distance, 
a the mean distance ; 
: Aa 
. Q= —A (energy of earth’s rotation) —3 mm p —. 
a 
Let hf denote the angular momentum of revolution, 
H the angular momentum of the earth’s rotation, 
then AH=—Ah; 
but pS Vp Va Vine; 
MV m--em' 
AR mm Vpn), Aa Va.ehe 
Vm+m' VIR eo 7 Aca 
IfS and N denote the components of the reaction of the disturbing forces exer- 
cised by the tidal protuberances, estimated tangential and normal (inwards) to the 
moon’s path, 
da 2a-7 8 
Sirens oilt Saud 
de ; N 2a (1-e*)(a-r) § 
ae R=er sin p a ad LAM 
where 7 is the moon’s radius-vector, 
¢ her longitude measured from apogee, 
v her velocity in the relative orbit. 
As both the coefficients of the disturbing forces in the last expression are small 
quantities of the order ¢, it follows that the second term in Ah is negligible with 
