ADDITIONAL NOTES. 51 



Mr. Airy (Tides and Waves, Art. 127) bases his demonstration of the theorem exclusively upon 

 tlu- principle of tin- con-ervation of ureas, remarking at the outset, " if the earth and sea were so 

 entirely disconnected that one of tin-in could revolve for any length of time with any velocity, in- 

 ava-iiiir or diminishing i" any manner, while the. other could revolve with any other velocity changing 

 in any other manner, we cmild pronounce, nothing as to tho effect of the fluctuation" (tidal) "upon 

 precession." 



A -phi-roidal nucleus wholly COM red by an ocean of uniform depth, suffering no resistance, does 

 not seem to me to lack much for fulfilling the above conditions. 



If ivliicitie* are gnu rated in the waters of the ocean by solar (or lunar) attraction, the centrifugal 

 forces duo to them might IK- looked to (though not alluded to by Laplace) as agents for transferring, 

 from the fluid to the nucleus, the precc:-Mon-protlucing couples due to the fluid mass, cs|tccially in 

 the ul>ove hvpnthctieul case. It will be found, huwcvcr, by reference to the expressions [2260], 

 that they give rise to no couple, and are, moreover, very minute. 



The motion which the displacements [-2200] [22G1] indicate is a slight oscillation of the axis of 

 the tluid envelope, moving as a solid, about the axis of the nucleus, the angular distance between 

 these axes being slight I v less than 2 seconds: it is, I presume, that which a non-rotating shell would 

 have were the attracting body, with constant distance and declination, to move, with angular velocity 

 n, in right ascension. In the case in hand it is the fluid shell which revolves and, suffering no 

 change of form, would he it.-df affected by its proper precessional couple to tho exclusion of the 

 oscillation above described. 



ADDITIONAL NOTES. 



NOTE TO PAGE 1 1. 



01 The process indicated is a more legitimate carrying out of the methods peculiar to this paper 

 than what follows in the text. The tangent of Mil' (35) may be (approx.) taken for the sine, and the 

 ciisim- taken constant at unity, as may also be the cos /'. 



From (32) we may calculate by developing and neglecting terms in which sin 1 /' enters 

 sin i = (1 cos* t)' = sin / sin /' cos / cos nj, 

 sin cos i = sin /cos I sin T cos2/cos nj. \ sin'T slnZI co&'nj 



Introducing these values in (38) and (39), and integrating we get expressions identical with 44 

 and 4.*i. except a (practically) immaterial difference in the coefficient oft in the first which becomes 

 1 siu'/' instead of 1 j sinT. 



NOTE TO PAGE 24. 



"' The foregoing interpretation of the symbolic integral in (7), adopted with hesitation from 

 authors cited, is based on assumed constancy of the angle <j> ; but this angle necessarily varies, 

 slowly indeed, but progressively, by the azimuthal motion measured by n sin x. The conditions for 

 the formation of a leminiscate are not, therefore, rigidly fulfilled. It will be found, however, taking 

 into account a complete excursion, that the slight increment which will enure to the moment of the 



quantity of motion, sin's -, from this cause on one side of the vertical, will be neutralized on the 



at 



other, in consequence of the opposing signs of cos f, in opposite azimuths; or, at least, the resultant 

 increment or decrement will lie. a quantity of the second order in minuteness, and hence, affecting 



only in the same degree the azimuthal motion JL 



at 



NOTE TO PAGE 39. 

 0) The differential attraction of the snn on any length dg of the rod, at distance x fr m '^ c enrth's 



(O a \ 



- 1 dg, r being sun's distance. Integrate from x = x to x = R 'the earth's 

 (r zr * / 

 radius). 



