24. REPORT—1874. 
respect to the first, and we may write 
Si Oe 
Vm+m' 2Va : 
. Q=—A (energy of earth’s rotation) + Vmspen' Vp AH; 
or, if I denote the moment of inertia of the earth, 
o her angular velocity of rotation, 
2 the mean angular velocity of the moon in her orbit, 
=—Io Ao+QlI Aw 
=—To Aw 1- o| ' 
o 
Tt follows that if w has the same sign as Q, not only is all the energy Q turned 
into heat drawn from the earth’s rotation, but, as a necessary concomitant, 
additional energy is transferred from the earth’s rotation to the store of potential 
and actual energy corresponding to the orbital motion of the system. 
It also follows that when © is, as in the actual case, very small compared to a, 
the energy so transferred bears a very small ratio to Q, and that the energy lost in 
the earth’s rotation is almost the exact equivalent of that consumed in friction. 
Let us now consider the case where the plane of the earth’s equator does not 
coincide with the plane of the orbit. 
Let G represent the resultant angular momentum of the system which will be 
fixed in magnitude and direction, 
6, © the angles which the planes of 2 and H make with the plane of G; 
then, since 
T?=G°+/?—2G .h cos 6, 
HAH=(h—G ¢os 6) Ah+sin 6.GhA 06; 
. _ mm'N a Aa —. 
“, HAH wn Mh 6 Cheondyigg HOWR in 0.8 
us mm'N wn a 
Aa j 
or AH= aa —cos (8+) 5; +s8in (+8) ae} ‘ 
Now if Z denote the component perpendicular to the plane of the orbit of the 
forces exercised by the protuberances on the moon, 
¥ the angle of elongation of the moon from the ascending node, 
= =C0s v5 . - 
where p= perpendicular from earth on tangent to lunar orbit : but 
1 da 2a-r S, 
Bis. taal Op iaaho baa 
it follows that A@ and = Aa in the expression for AH bear in general a finite 
but, in the absence of further information about the position of the protuberances, 
unknown ratio to each other. 
Let the ratio of the first to the second be denoted by X; then 
NV m-+m' 
hes £0 Aes, ai BOLO TI) +, 
Q= To Ao 1 So rae 
AHe — 27m VE jit (6+8) al cos (6+8) pase: 
