Mr. D. D. Heath on the Problem of Sea-levels. 37 



relatively displaced from one epoch to the other cow instead of w. 

 Hence his formula -^ ~- 



1 — (OW 



It does not belong to me to praise the neatness of Professor 

 Thomson's work ; bat I must remark upon his observation that 

 the first term in Laplace's series "always expresses the essence of 

 the result/' and also upon the applicability of his solution to 

 the actual geological question. 



He cannot mean that in every case where Laplace's methods 

 can be used the first term is of a higher order of magnitude than 

 the sum of all the subsequent terms. I suppose, therefore, that 

 he must mean to lay it down that in the passage from one set 

 of disturbing forces to an approximately equal and opposite state 

 — as from a southerly to a northerly epoch — this will be the case. 



I appeal to my calculation of a special case to show that this 

 is not so. With a uniform cap extending to N.P.D. 30°, I find 

 the total displacement at the pole is 1078 + 206, or 1284, of 

 which the part due to the shifting of the centre is 750. 



And if we look to the rationale of the thing, it is evident that 

 the peculiar result he has come to implies that, with that special 

 law of thickness and extension of the capping, the form assumed 

 by the surface is symmetrical in the two hemispheres, or, which 

 is the same thing, that its equation, referred to an origin near 

 the centre of figure (such as the centre of the nucleus), is of the 

 form r — fl = a|« 1 Q 1 + a 2 Q 2 + a 4 Q 4 + &C.J-, with no odd terms 



after the first : and this is easily verified*. But when the form 

 is u-nsyra metrical and other odd terms appear in the equation, 

 then the change of level, referred to the same origin, in passing 

 from this state to its opposite, is due to the shifting both of the 

 contour-line and of the centre of figure ; and the relative amounts 

 at the pole are as the sum of the coefficients of all the odd terms 

 after the first to that of the first. 



* Referring to my March paper, it will be seen that when the density 

 varies as //, u n a will be a multiple of J Q'^/x' dp' between proper limits. Now 



d J dQ n I 



Whence, multiplying by \x and integrating by parts, it will be found (when 

 n > 1 ) that 



(n-\)(n + 2)fQ n ^=C-(l-^)(^-Q n ). 



Taken from any given latitude to the pole, this becomes 



(W.-)(ff-Q.)» 



and when we start from the equator it reduces to — Q«, which vanishes 

 when n is odd. 



