190 



to make it coincide in period with that in the orbit, supposing that 

 there had ever been a time when they did not do so. 

 The general results arrived at may be thus stated : — 



(1) . That in all bodies which are perfectly symmetrical with respect 

 to the three planes containing their principal axes of rotation, such as 

 an ellipsoid (not one of revolution), there is no permanent change pro- 

 duced; nor in any in which the moments of inertia about the two 

 principal axes which are perpendicular to that about which it is revolv- 

 ing are equal ; I think there is a permanent change whenever the three 

 moments are unequal, though the body may be perfectly symmetrical 

 with respect to each axis ; 



(2) . That in other cases there is in general a permanent change 

 produced ; 



(3) . That in the case of the earth disturbed by the lunar moon, the 

 change produced will be so very small as to account for a very minute 

 fraction of the whole amount required to explain the phenomena above 

 alluded to ; 



(4) . That one condition under which there will be no permanent 

 variation, is when the time of rotation nearly or exactly coincides with 

 that in the orbit ; but this is only one out of several other such rela- 

 tions as might exist; just as there are always several positions in 

 which a body might remain in statical equilibrium ; and that in some 

 cases, though not in all, the forces are such as to produce the relation 

 above spoken of; and, lastly, 



(5) . That the effects are enormously more rapid in the case of a sa- 

 tellite described by its primary than vice versa. 



I have supposed, in treating the question, only one disturbing force 

 to be acting upon the body, and also that its orbit is a fixed plane ; 

 neither of which, especially for the earth or moon, is strictly the case, 

 but will be sufficiently near the truth for the present purpose ; also, I 

 have supposed the body to be entirely solid, instead of being partially 

 covered with a thin layer of fluid. Mr. Airy, however, is said to have 

 examined the effect of the tidal wave, which it was supposed might by 

 its position, &c, produce some retardation upon the earth's motion, and 

 has found it to be insensible. 



Result (4.) has been spoken of as only an approximate one — indeed, 

 to pretend to extract anything more out of differential equations which 

 can only be solved by successive approximations, as is the case in the pre- 

 sent instance, would seem almost to amount to a contradiction of terms. 



2. The differential equations of motion are 



SI A "A ((x- a,) 2 t {y - Vif *•■('- *.) ! ) T 

 dio 2 A - C 1 xz x - zx x 



dt 



+ 



{ x - x? + y - y x 2 + z - Z\ 



> A. 



~dt 



B-A 

 C 



{(x - x; 2 + y - y 2 + % - ss! 2 j 



