200 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1949 
motion of the moon. In the development of the theory, the gravita- 
tional effects which have been neglected are far too small to amount 
to anything like the discordances which are observed. In more 
recent years it has been proved that there are similar fluctuations in 
the motions of Mercury, Venus, and the sun; but for these bodies 
the effects are much smaller than for the moon because their mean 
motions are much less rapid. It was the comparative smallness of 
the effects for these bodies which made their detection difficult. So 
there are, in effect, four clocks which agree together and one clock, 
our earth, which differs from the other four. The natural conclusion 
is that it is the earth which is at fault and that the length of the day, 
which has been adopted as the unit of time and assumed to be invar- 
iable, is actually subject to small variations. 
The changes in the length of the day are found, from the analysis 
of the observational data, to be of two different kinds. There is a 
slow progressive increase in length, of the order of 1 millisecond in 
the length of the day in the course of a century. This progressive 
increase is caused by tidal friction, more particularly in the shallow 
sea; it acts as a brake on the earth. Though so small in amount, the 
effect on the mean longitudes of the moon and the planets increases 
with the square of the time and is large enough to make the position 
of the moon 20 centuries ago, if computed from its present motion in 
longitude, very considerably in error. The effect was actually first 
detected in 1679 by Hailey from the early observations of eclipses. 
Superposed on the progressive increase of length there are also irregular 
changes, the day sometimes increasing in length and sometimes de- 
creasing; these changes cannot be attributed to tidal friction, because 
frictional effects can cause only a slowing down and never a speeding 
up in the earth’s rotation. These changes are due to changes in the 
earth’s moment of inertia and could be accounted for quantitatively 
if the earth expanded or contracted slightly by 4 or 5 inches. 
There is one essential difference between the two phenomena. 
A change in the moment of inertia of the earth is something that con- 
cerns the earth alone. The apparent displacements of all the other 
bodies are strictly proportional to their mean motions. But tidal 
friction is something that concerns the earth and the moon jointly; 
the total angular momentum of the earth-moon system is conserved, 
but there is interaction between the earth and the moon. The appar- 
ent displacements of Mercury, Venus, and the sun will again be 
proportional to their mean motions but the same will not hold for 
the moon; its displacement will not have the same ratio to its mean 
motion. It is this difference in the case of the moon which makes it 
possible to separate the two effects of tidal friction and of change 
of the moment of inertia of the earth. 
