Mr. D. D. Heath on the Problem of Sea-levels. 39 



Still, even when corrected, we are thus presented with a dis- 

 continuous expression of the attraction exerted by the disturbing 

 force on points of the ocean's surface. And this seems to me a 

 formidable obstacle to dealing with the problem in the only way 

 I think admissible, and which the Archdeacon himself at first 

 propounds (Figure of the Earth, p. 155), viz. by endeavouring 

 to satisfy the equation (in which I leave out the term for the 

 centrifugal force as immaterial) 



f: 



= V+ £\Jdu = const., 

 p J 



where V is the internal potential, and J \Jdu that due to the dis- 

 turbing force. 



I conceive it would be necessary to assume a form of surface, 

 with indeterminate coefficients, having the same discontinuous 

 passage from one hemisphere to the other, thence to calculate V 

 to the same order of small magnitudes as \ Vdu, then to deter- 

 mine the coefficients so as to satisfy the equation at all parts of 

 the surface, and finally to determine the constant of integra- 

 tion so as to ensure the permanence of the mass of water in the 

 disturbed and undisturbed spheroid. 



But Archdeacon Pratt proceeds to assume for V its value in 

 the undisturbed spheroid, — a liberty which may be justifiable in 

 the case of an isolated sea like the Caspian, in which any sensible 

 effect of an overhanging table-land or mountain-range would 

 probably overwhelm that due to any possible change of surface 

 of the sea, but which seems to me wholly illegitimate when we 

 are dealing with an ocean covering three-fourths of the globe, 

 which can and will adapt itself to the new forces. 



But having thus obtained, however wrongly, his equation to 

 the disturbed surface, viz. 



he does not proceed to determine the constant of integration so 

 that the mass may remain unaltered, but he goes through a cal- 

 culation which is equivalent to the following : — 

 If E be the radius of tbe undisturbed surface 



whence, subtracting and transforming, 



r — R= -n-^Vdu= y f Vdu, or- £\Jdi, nearly. 



