﻿414 Rev. J. J. Blake on 



necessary to assume the equality of the two sides to the same 

 degree of approximation as there are layers in the crust/' by 

 saying that there is no necessity for such an equality in 

 nature. Just so — that is my objection — that the equality has 

 to be assumed for the sake of the mathematics. 



These minor objections, however, and their replies are of 

 comparatively little consequence; and I merely mentioned 

 them as slight indications that should have shown that some- 

 thing was amiss. The main statement, which Mr. Fisher 

 simply repeats, is to the following effect : — If 



(S„,(T0-2„(r'O)/W+(S M (rf)-S„(^))^+&c.=0 . (A) 



for any number of assumed values of 0, then must 

 t m (rt) =2 w (t¥) "J 

 Z m (Tt 2 )=% n (T't'*)\ (B) 



&c. = &c. J 



the t's denoting densities and the tf's thicknesses. 



That is his fundamental proposition, and what I say is this : — 

 that equations (B) do not follow from (A), and that if they 

 were true, they would be of no value. 



Even the converse proposition, that if equations (B) are 

 satisfied the series (A) is zero, is only true if the series is con- 

 vergent; but to deduce (B) from the series (A) being zero it is 

 necessary to show, first, that it is convergent ; secondly, that 

 none of the functions f(0), <l>{0), &c, are zero for all values of 

 0, and particularly that it is possible to make <j>(0), ty(0), &c. 

 zero without making/(0) zero at the same time. This is what I 

 mean by " independent." Mr. Fisher replies to this, that they 

 are not independent, which is exactly my objection, because he 

 has to assume them, in his work, to be so. As a matter of fact, 

 if we take Mr. Fisher's values of/(#), <f>(0), ty(0), as correct, 

 each is a linear function of the other two and all vanish 

 together for the same value of 0, so that the coefficient of any 

 one may be distributed amongst the others. Moreover, the only 



value of which makes them zero is given by sin- = — 1, 



which is an impossible value in the problem dealt with, 



while the occurrence of sin- in the denominator of <j>(0) 



L 



and ^fr(0) shows that the series may well be divergent for 

 small values of 0, and, indeed, cannot be a true expression for 

 the attraction when 0=0. The true expression is in fact a 



function of two independent variables t and sin - ; and if we 



