60 



ADDENDUM. 



ADDENDUM TO NOTE I, PAGE 38. 



THE apparent antagonism between the theorem of the text and that of Laplace suggests a few 

 additional words. The theorem of Laplace is that " in whatever manner the waters of the ocean 

 act upon the earth, either by their attraction, their pressure, their friction, or by the various resist- 

 ances which they suffer, they communicate to the axis of the earth a motion which is very nearly 

 equal to that it would acquire from the action of the sun and moon upon the sea, if it form a solid 

 mass with the earth." (Me'c. Cel., Bowditch [3345].) 



The theorem is demonstrated in two distinct, quite different, manners. The last demonstration is 

 founded upon the principle of the "conservation of areas;" aud as the result of this demonstration 

 the proposition is stated in the above quoted words. 



The first demonstration is purely analytical, and, after stating that " this fluid" (i. e. of the ocean) 

 "acts upon the terrestrial spheroid by its pressure and by its attraction," Laplace proceeds to find 

 the analytical expressions for the precession and nutation-producing couples due to this pressure and 

 to this attraction as they are modified by the attraction of the sun and moon upon the fluid. He 

 then proceeds to calculate these couples for the material substance of the ocean, considered as rigidly 

 connected (or forming a solid mass) with the earth. He finds the couples, so calculated, respectively, 

 identical in the two cases, and epitomizes the result as follows : " the phenomena of the precession 

 of the equinoxes and the nutation of the earth's axis are exactly the same as if the sea form a solid 

 mass with the spheroid which it covers." [3281.] 



But this demonstration is limited by the assumption that " the sea wholly covers the terrestrial 

 spheroid or nucleus, that is of a regular depth, and suffers no resistance from the nucleus;" and both 

 demonstrations imply an ocean of (relatively) small depth. 



Under the last mentioned treatment of the subject the proposition of Laplace and that which I 

 demonstrate are but the extreme phases exhibited by the solution of a problem, according as the 

 datum be that the depth of the sea is minute (in which case its entire precession-producing couple, 

 not effectively exerted upon its own mass, is almost wholly transferred to the solid nucleus) ; or 

 that the nucleus is very small, in which case the lost precession-producing couple of the fluid is but 

 in small part transferred to the nucleus or wholly disappears with the vanishing of the latter. 



I think, however, that the last-mentioned (first in point of order) demonstration of Laplace is not 

 as general as the language quoted [3287] would indicate. The omission of variations of the radius- 

 vector, R, in all the integrations gives rise to errors which do not seem to ine to be identical in the 

 two processes by which the couples are calculated, when the variation of depth is very small. 



An apt illustration of the above remarks is derived from the supposition as admissible as any 

 other that, the depth, y, is constant. In this case, whatever be the ellipticily of- the solid nucleus, 

 the value of y (the height of the diurnal or precession-affecting tide, see [2253] [3333] and also p. 36) 

 is zero, and, of course, the couples [3272] and [3273] become zero as will be found by performing 

 the integrations in those equations. So, of course, do expressions [3284] and [3285] become zero, 

 with y made constant. But these last should not be zero except for the case of sphericity of the 

 nucleus. 1 The expressions do not seem to me capable of snstaining the inference which I have 

 quoted [3287], when the depth of the sea is uniform; a case which most naturally presents itself to 

 the mind. 



1 A shell of slight internal elliptioity and small uniform thickness has a precessional coefficient of four-fifths the 

 value of that of a shell bounded by surfaces of equal elMpticity. Hence in general the variations of R (or the 

 ellipticity) produce effects insensible compared to those of the depth. 



