beneath continents and oceans compared. 115 



is less than that of the lower, consisting of plutonic rocks. Hence 

 (fa — fa) will be positive. So also the density of the substratum 

 being greater than that of the crust, (<x — fa) will be positive. 



Hence, changing the signs of all factors which are evidently 

 negative into positive, and in (5) changing provisionally (d — 8) 

 into (8 — d), to make it agree with (1), we shall have changed the 

 signs of an odd number of factors in each except in (4), viz. in 

 (1) five, in (2) three, in (3) three, in (5) five, in (6) one. All 

 the factors in (4) are obviously positive except the density factor 

 (</ — cr). We are therefore obliged to change it provisionally into 

 (a — a'). Then since the several products, being equal, had the 

 same sign originally, they will all have the same sign now as one 

 another ; and what that sign will be will depend upon the sign of 

 (8 — Cj) which we must seek to discover. 



Our five equations will now be 



(p 1 - f ,)8fa(fa + fa)(d-8)(8-c 1 )(c 1 + c 2 -8) (1) 



= (pi - p-i)(8 + fa) fafa (d — 8- fa)(8 + fa- d)(Ci + c 2 — 8-fa). . .(2) 

 = (a — p 2 ) (8 + fa + fa) (fa + fa) fa(d—8 — fa- fa) 



(8 + fa + fa- Cj)(ci + c 2 - 8 - fa - fa) . . .(3) 

 = (a - *') d (d - 8) (d - 8 - fa) (d-8-fa- k. 2 ) 



(d — d) (d — c 1 — c 2 ) . . .(4) 

 = (</> 2 - fa) d (8 - d) (8 - c 1 + fa) (8-c 1 +k 1 + k 2 ) (d - c 3 ) c 2 . . .(5) 

 = (o- - fa) (c x + c 2 ) (d + c 2 - 8) (d + c 2 - 8 - fa) 



(cj + c 2 - 8 - fa - fa) {d-c l - c 2 ) c 2 . . .(6). 



In order to determine the sign of (8 — d), that is whether the 

 depth of the ocean or the thickness of the upper layer of the 

 continental crust is the greater, consider the products (1) and (5), 

 and we have the equation 



(p! - /x) 8 (d - 8) (8 - d) (d + c 2 - 8) fa (fa + k 2 ) 

 = (fa - fa) d(S - c x )(d - d)(ci + c 2 - c 1 )(k 1 + 8- c 1 )(k 1 + k 2 + 8 -d), 



and each side of this equation must have the same sign. It is 

 obvious that the factor (8 — d) will not affect the question. We 

 therefore divide it out, and it remains to find the relative magni- 

 tude of 8 and c u such that 



( Pl -fi)B(d-S) (d + c 2 - 8) fa (fa + fa) 

 may have the same sign as 



(#2 - fa) Ci (d - c,) (d + c, - d) (fa + 8 - d) (fa + fa + 8 - d). 

 Write a and /3 for the factors which do not involve fa and fa, 

 and we have 



afa (fa + fa) = /3(fa + 8- d) (fa + fa + 8 - d) (A), 



