AT THE SURFACE OF THE EARTH. 689 



in consequence of the fresh distribution of attracting matter. The surface would thus preserve traces 

 of its original irregularity. A repetition of the same process would give a surface still more regular, 

 and so on indefinitely. It is easy to see the general nature of the correction which still remains. 

 Where a small island was cut off, there was previously no material elevation of the sea-level, and 

 therefore the surface obtained by cutting off the island and replacing the surrounding sea by land 

 will be very nearly a surface of equilibrium, except in so far as that may be prevented by alterations 

 which take place on a large scale. But where a continent is cut off there was a considerable elevation 

 in the sea-level, and therefore the surface which is left will be materially raised above the surface of 

 equilibrium which most nearly represents the earth's surface in its altered state. Hence the general 

 effect of the additional correction will be to increase that part of g' which is due to causes which act 

 on a larger scale, and to leave nearly unaffected that part which is due to causes which are more 

 local. 



The form of the surface of equilibrium which would be finally obtained depends on the new 

 distribution of matter, and conversely, the necessary distribution of matter depends on the form of 

 the final surface. The determination of this surface is however easy by means of Laplace's analysis. 



26. Conceive the sea replaced by solid matter, of density cr, having a height from the bottom 

 upwards which is to the depth of the sea as 1 to <r. Let h be the height of the land above the actual 

 sea-level, h being negative in the case of the sea, and equal to the depth of the sea multiplied by 

 1 — CT"'. Let X be the unknown thickness of the stratum which must be removed in order to leave 

 the surface a surface of equilibrium, and suppose the mean value of x to be zero, so that on the whole 

 matter is neither added nor taken away. The surface of equilibrium which would be thus obtained 

 is evidently the same as that which would be formed if the elevated portions of the irregular surface 

 were to become fluid and to run down. 



Let V be the potential of the whole mass in its first state, F, the potential of the 

 stratum removed. The removal of this stratum will depress the surface of equilibrium by the 

 space G"'r, ; and the condition to be satisfied is, that this new surface of equilibrium, or else a 

 surface of equilibrium belonging to the same system, and therefore derived from the former by 

 further diminishing the radius vector by the small quantity c', shall coincide with the actual 

 surface. We must therefore have 



G-'F, 4 c' = .r-A (S^^) 



Let h and x be expanded in series of Laplace's coefficients h„ + h^ +... and Xd + x^ +... Then 

 the value of V, at the surface will be obtained from either of equations (28) by replacing I by ax 

 and putting r = a. We liave therefore 



F, = 4 7ro-a(.r„ + i.r, + }x„ + ...) (34.) 



After substituting in (33) the preceding expressions for F„ h, and x, we must equate to zero 

 Laplace's coefficients of the same order. The condition that x^ = may be satisfied by means 

 of the constant c', and we shall have 



G~' ■ iwaa = Xi — hi, 



2i + l 



which gives, on replacing G Kiiraa by its equivalent — , 



(ii+jlf „J,+^ 1,,, (.3.,) 



' (2i+ \)p-3<T ^ I (2«+ l)p-:ia\ 



We see that for terms of a high order .r, is very nearly equal to //„ but for terms of a low order, 



whereby the distribution of land and sea would be expressed as to its broad features, .r^ is sensibly 



greater tliaii //,. 



