44-47] . The Tidal Problem 43 



45. It will be understood that the foregoing discussion of stability has 

 been concerned only with secular stability, this being the only kind of stability 

 which is of interest in problems of cosmogony. The conditions of ordinary 

 stability are quite different ; for instance it has been shewn by G. H. Bryan * 

 that Maclaurin's spheroid remains ordinarily stable until its eccentricity is 

 given by e = "9529. 



II. TIDALLY DISTORTED ELLIPSOIDS 



46. We now pass to a problem in which the distinction between secular 

 and ordinary stability disappears. 



A distant heavy mass will raise tides in a spherical mass of fluid, so that 

 the fluid assumes the shape of a prolate spheroid. As the heavy mass 

 approaches, the eccentricity of this spheroid will increase, and the question 

 arises whether the spheroidal form remains stable no matter how great its 

 eccentricity. The bearing of this problem on the planetesimal theory and 

 other tidal-distortion theories is obvious. 



47. Suppose that a mass M of fluid which we shall call the primary, is 

 acted on by tidal forces originating from a second mass M', which we shall 

 call the secondary. Let us at first suppose that the mass M ' of the secondary 

 is collected in a point, this being of course a legitimate approximation if the 

 secondary is at a great distance from the primary. 



Let the centre of gravity of the primary be taken for origin, and let the 

 secondary be at a distance R, its spherical polar coordinates (r, 6, <) being 

 supposed to be R, 0, 0. The tide -generating potential at the point r, 6, <f> 

 will be 



M' M' M'rcosO M'r* D M'r* D 



L-TBT + - -+ -TiT P 2 ( COS 0) + ~B7 A(COS<9) + ... 



(73). 



The first term on the right M'/R is a constant and so gives rise to no 

 forces on the primary mass. The second term gives a uniform field of force 

 of intensity M'/R 2 , which produces the Newtonian acceleration M'/R 2 in the 

 primary. We can neutralise this term by supposing the axes of reference to 

 move with an acceleration M'/R 2 ; the centre of gravity of the primary will 

 then always remain at the origin. 1 



We are left with a tide-generating potential 



^~r ^2 (cos 6) + i P 3 (cos 6) + (74). 



* Phil. Trans. A 190 (1888), p. 187. 



