404 
Proceedings of the Royal Society 
for the proportionate gain per second. There are 31*5 million 
seconds in a year, and 3150 in a century. Hence the ratio to the 
earth’s present angular velocity, of the gain per second, amounts to 
1-73 x 10 9 . 
To interpret the result, suppose two chronometers, A and B, to he 
kept going for a century, according to the following conditions : — 
Chronometer A to he an absolutely perfect timekeeper, and to 
be regulated to sidereal time at the beginning of the century, in the 
usual manner, by astronomical observation. 
Chronometer B to be kept constantly regulated to sidereal time 
by astronomical observations from day to day, and from year to 
year, during the century. 
At the end of the century B will be found to be gaining on A 
to the amount of 1 *73 x 10 “ 9 of a second per second This rate of 
gain has been uniformly acquired ; and, therefore, on the average of 
the century, B has been going faster than A, at the rate of 
*86 x 10 ~ 9 of a second per second. Hence, in the whole century (or 
3T6 x 10 9 sidereal seconds), B has gained on A to the extent of 
2*7 seconds. 
In reality a tenfold greater difference, in the opposite direction, 
would be observed between the two chronometers. Adams, from 
his correction of Laplace’s dynamical investigation of the accelera- 
tion of the moon’s mean motion, produced by the sun’s attraction, 
found that our supposed chronometer B, regulated to sidereal time, 
would be 22 seconds behind the perfect chronometer A at the end 
of a century. (See Thomson and Taifs Natural Philosophy , 1st 
ed., § 830 ; or 2nd ed., vol. i. part 1, § 405.) The retardation 
of the earth’s rotation thus definitively specified, which may be 
regarded as a well-established result of observation and theory, 
received from Delaunay what we cannot doubt to be its -true 
explanation, — retardation by tidal friction. The preceding 
formulas, with the proper change of data, may be readily modified 
to show the tidal retardation instead of the thermodynamic accelera- 
tion. Thus if we go back to fig. 1, and suppose the spheroidal 
layer to be water, instead of the earth’s atmosphere, and take 100 
cms. as the excess of the greatest above the least semi-diameter, 
we have what we may fairly assume to be a not improbable 
