382 PROCEEDINGS OF THE AMERICAN ACADEMY 



cause. Although we cannot determine the exact amount of effect aris- 

 ing from tidal action on account of the uncertainty of the data upon 

 wliich the computations are based, yet it may be shown that it requires 

 no extravagant and unreasonable liypothesis with regard to the magni- 

 tude of the tidal wave, and its retardation by friction, to produce an 

 apparent secular acceleration of 6", after making due allowance of the 

 eflPect of tidal action for counteracting the effect arising from any prob- 

 able amount of contraction of the earth's volume. If upon the hy- 

 pothesis of a tidal wave two feet in height, and displaced 30° by fric- 

 tion, the effect of tidal action is to cause an apparent secular accelera- 

 tion of 84", then, since the effect is as the sine of double the angle of 

 displacement, an acceleration of 6" would be obtained from a wave of 

 the same height displaced only about 2°, that is, retarded only about 

 eight minutes, by friction. This is upon the hypothesis of no contrac- 

 tion of the earth's volume. According to the investigations of Fou- 

 rier the loss of heat sustained by the earth by conduction, in a century, 

 would only be sufficient to cause a diminution of the earth's radius to 

 the amount of about seven millimetres, (nearly one third of an inch) 

 the effect of which upon the earth's rotation would be counteracted by 

 an additional retardation by friction of the tidal wave of only about 

 two minutes. The retarding pressure assumed by Mayer corresponds 

 with the effect of a tidal wave two feet high delayed about 1| hours 

 by friction, but from this assumption he obtains a result which is 

 twenty-six times too great for Fourier's theory of the loss of heat sus- 

 tained by the earth. But he thinks that the loss of heat by simple 

 conduction taken into account in Fourier's investigations, may be but a 

 small part of the whole amount, and enumerates several other ways in 

 which the earth loses heat. Although the loss of heat may be consid- 

 erably greater than that which would result from Fourier's theory, yet 

 I think the whole amount of loss cannot be so great as to require any 

 unreasonable amount of retardation of the tidal wave by friction, in order 

 that the effect may be great enough to counteract the effect of the con- 

 traction of the earth's volume, and also to retard the time of the earth's 

 rotation enough to cause an apparent secular acceleration of 6" per 

 century in the moon's mean motion, and that the hypothesis in my 

 paper of a displacement of 30°, or retardation of two hours, and also 

 Mayer's hypothesis of the amount of tidal pi'essure, which corresponds 

 to a retardation of about 1^ hours, are both much too great to be rea- 

 sonable hypotheses, or to be necessary to account for the balance of the 



