74 Air Love, On Sir William Thomsons estimate [April 28, 



three components parallel to the coordinate axes, and each of these 

 is equated to zero. The result is the determination of all the un- 

 known harmonics occurring in the complementary functions, i.e. 

 the complete solution of the problem, and in particular an ex- 

 pression is obtained for the amount of the harmonic inequality. 



It is of some importance to notice the way in which gravity 

 rigidity, and compressibility, occur in the result, and I shall for 

 simplicity of statement here limit the expression of the results to 

 two cases. In the first, which is that considered by Thomson and 

 Tait, the matter is supposed incompressible. In the second the 

 matter is supposed compressible as well as rigid in such a way that 

 the constants m and n are connected by the relation m = 2n, equi- 

 valent to the relation 3k = 5n above referred to. In both the 

 density is supposed equal to the earth's mean density, and the dis- 

 turbing forces derivable from a potential, which is a spherical solid 

 harmonic of order 2, say W 2 . This is the case for tidal disturbing 

 forces. Then in both cases the amount of the harmonic inequality 

 is expressible in the form eWJg where g is the value of gravity at 

 the surface, and the number e is a rational function of a certain 

 number ^ such that (3^) _i is the ratio of the velocity of waves of 

 distortion in the material to the velocity due to falling through a 

 height equal to half the radius of the sphere under gravity kept 

 constant and equal to that at the surface. In the first case I find 



15fr (K . 



agreeing with Thomson and Tait's result, and in the second case 



S 3356500 + 863100S + 55485S- 8 ( , 



6 ~ 70 + 9S- ' 53900 + 27160* + 2601*' '"^' 



The value of * is zero when the matter is perfectly rigid, about 

 | when the rigidity is that of steel, about 5 when the rigidity is 

 that of glass, and infinite when there is no rigidity at all. 



We may regard e as an ordinate and * as an abscissa and 

 trace the curves (b) and (c). The curve (6) is a rectangular hy- 

 perbola, and the asymptote e = constant gives the limiting value f 

 of e when the rigidity vanishes. The branch of the curve (c) which 

 passes through the origin lies very near to the corresponding branch 

 of the curve (b) for all positive values of S-. It has an asymptote 

 giving the limiting value of e something less than f. For all values 

 of S- lying between S- = and * = 5, the curve (6) lies below the 

 curve (c) but the distance is very minute. For some value greater 

 than S- = 5 the curve (b) crosses the curve (c) and the value of e 

 given by supposing the matter incompressible is greater than that 

 given by supposing 3k = 5n. The value of e for steel (* = §) is 

 about '803 as given by the curve (b) and about '856 as given by (c), 



