LAMBERT: CONSTITUTION OF THE EARTH 135 



constants of the earth is difficult and leads to such intricate 

 mathematics that we have to be content with very simple hypo- 

 theses about the elastic constants and with rough approximations. 

 The forces that deform the earth have a potential variable 

 with the time and place on the earth. Suppose this potential 

 to be expanded in a series of spherical harmonics, and let us con- 

 sider only that portion due to the spherical harmonic of given 

 order, which we shall denote by W. For the tidal deformation 

 TT'' is of order two. At the surface of the earth the force acts 

 against gravity (g) so that the vertical displacement of a single 

 particle would be W/g. This quantity measures the vertical 

 displacement of the equipotential surfaces due to normal gravity. 

 If the earth were covered with a non- attracting fluid whose sole 

 function would be to make manifest the instantaneous form of 

 the surface of equilibrium by conforming immediately to the 

 forces, W/g would be the height of the tide in this fluid at the 

 time and place in question, and the quantity W/g may be called 

 the "equilibrium tide," due to the forces given by W. But the 

 particles disturbed by the action of the forces are not weightless 

 and non-attracting, which fact gives rise to new forces of the same 

 type as these represented by IF, but of different magnitude, the 

 identity of type being, of course, a characteristic of the spherical 

 harmonic expansion. These new forces cause further displace- 

 ment which again cause new forces and so on indefinitely. The 

 aggregate effect is, however, finite; the force called into being 

 by a displacement of attracting matter is less than the force that 

 produced the displacement. When equilibrium is attained, — 

 we suppose that the period of W is long compared with the time 

 necessary to adjust conditions to momentary equilibrium — the 

 total vertical displacement of the equipotential surfaces at any 

 point is no longer W/g, but a quantity of the same kind, hW/g; 

 this defines h. h is a quantity such that hW/g measures the 

 actual displacement when the displacement due to the equilib- 

 rium tide alone is W/g. Further, owing to the rearrangement 

 of attracting matter, the potential of the force acting is no longer 

 W alone, but a larger quantity (i + k) W; this defines k. k is 



