16 INTERNAL STRUCTURE OF THE EARTU. 



APPENDIX I. 



{Note to Page 10.) 



It is curious that the demonstration of Hopkins, Sections 10-13 (Phil. Trans., 

 I83y), taken in all the abstractness (not fully realized on first reading) which really 

 belongs to it, involves a mechanical paradox, a real analogue of that in the last 

 clause of § 25 of Sir W. Thomson's " Rigidity of the Earth." 



The planes of fluid diurnal rotation being slightly tilted by the displacement of 

 the shell, we have the " repelling line" again, but one of which, the radiating repul- 

 sion, is exerted in parallel planes slightly oblique to the line in which lies their origin. 

 Now in all its generality Mr. Hopkins' demonstration is this. A fluid (the rotation 

 of which is like that of the § 23 of the "Rigidity of the Earth" "stopped" and 

 repulsive force substituted for the rotary motion) inclosed in an envelope of any 

 form., subjected to the diverted force just desci'ibed, will be acted on at each point 

 by a rotational force for which an analytical expression is found. The demonstra- 

 tion flows from the received laws of "equilibrium" of inclosed fluids, and here we 

 see again the anomalous results to wliich the substitution of s, force for tlie motion 

 to which it is due may lead. If the fluid takes up this rotation while everything 

 else remains unchanged, the action is " perpetual ;" and indefinite, thus induced, 

 rotational velocities ensue. 



But in reality the rotational force so found is reaction of the envelope against 

 the reverse rotational motion impressed on it by the centrifugal force of the fiiiid. 

 That reaction jtushes the planes of rotation into their deviation, and, as if by action 

 of a spring, causing the incidental subsequent rotation " as a solid." In fict the 

 fluid cannot yield otherwise than by this yielding of rotational planes, as such, and 

 at tlie same time preserve intact its diurnal rotation. 



Overlooking this relation between the impressed rotations on shell and fluid, 

 which must be in the inverse ratio of their respective moments of inertia (since the 

 "action" and "reaction" to which they are due are equal) — a relation which is 

 actually realized (though not recognized) in Mv. Hopkins' results for homogeneous 

 shell and fluid — he foiled to detect the error of his integrated pressures (the inte- 

 grals of J^ Rdr separately computed, pp. 199-201, Phil. Trans., 1840) exerted on 



the shell by the heterogeneous fluid — unrecognized errors on which (as I have 

 pointed out in Addendum to " Problems," etc.), his conclusions as to the preces- 

 sion of heterogeneousness internally fluid earth are based. The error of estima- 

 ting as pressure on the shell the total of the above integrals, lies in the fact that a 

 pai-t is consumed in maintaining special configurations in the strata of equal 

 density; configurations which Hopkins liimself (iu another connection) investigates 

 and defines. 



