﻿" Densities in the Earth's Crust. 



415 



attempt to expand it in powers of these the coefficients arc 

 found to be indeterminate. 



But supposing equations (B) were true, what do they 

 moan ? 



As t t 7 &c. may be positive or negative, we can write them 



T\h +T 2 t 2 +T 3 * 3 + T 4 £ 4 . . . = 



^l 3 + T 2 ^ 3 + T 3 ^3 3 + T^ 4 3 

 ^1 4 + T 2 ^+T 3 ^ 3 4 + T 4 f 4 4 



. = 

 . = 

 . = 



Then, unless t 2 t 2 t 3 &c. are all zero, we have 



h h h h 



/ 2 / 3 /2 / 2 



ij t-2 c 3 c 4 



^1 ^2 ^3 ^4 



ti t 2 t 3 t± 



= 



or *! * 2 M 4 • • • (*i-'a) («i-^)A-0(«i-«i) • • • = 0. 

 Hence one of these factors must be zero, and therefore by 

 symmetry all in the same group ; that is, either all the thick- 

 nesses of the layers are zero, or all are equal; and in the 



latter case we get 



Tl + T 2 + T 3 + T 4 . . . = 0, 



or the sum of the densities in one cap is equal to that in 

 other. 



Mr. Fisher nearly came to this point when he supposed his 

 a quantities t\t 2i &c. to be connected by n — 1 equations. If 

 he had only tried n equations instead he would have come to 

 this same result. It may be noted that the first n equations 

 (B) give us t 1 = t 2 = ts 1 &c, and hence any number of such 

 equations will be satisfied, for they all reduce to 



Tl + T 2 + T 3 + T 4- • • =0. 



Now since the supposed layers of each cap overlap — for I 

 admit that my second objection is answered and that this is 

 assumed in the working — and they are all equal, the division 

 of the cap into coincident layers of different density is quite 

 arbitrary, provided the sum of the densities of the parts is 

 the same in the two caps. Equations (B) mean, therefore, 

 that the two caps are identical in all respects. In other 

 words, we get to the following curious result, that Mr. Fisher's 

 mathematical argument, if assumed to be correct, would show 

 that identical attractions can only be produced by identical 



