beneath continents and oceans compared. 117 



We have proved that 8 — c 1 is positive, if 



Pi-P 



/(*) >/(*). 



<f>2~ </>l 



To discover under what conditions this is the case, we must 

 assume some probable values as a working hypothesis. p 1 being 

 the density of some kind of rock, and p, the density of water, we 

 may put p 1 — p,= 1*6. If then we assume <r = 3, for the density of 

 the substratum under the land, and 0'2 as the difference between 

 the densities of the two layers of the continental crust, we have 



Pi- ft = 1'6 o 

 fc-fc 0-2 * 



Let us also assume the entire thickness, Cj + c 2 , of the conti- 

 nental crust to be 25 miles, and let us consider the state of affairs 

 to the depth of 100 miles, so that d = 100. 



Then tracing the curve 



f(x) = x (25 - a?) (100 - a;), 



we observe that it cuts the axis of x at the origin, and at x— 25, 

 and x = 100, 



f(x) is a maximum when x= 11 "62, 



and the value of the function then becomes 13741. If we now 

 seek the value of x which makes 8f(x) to have the same value as 

 the maximum value of f (x), i.e. to solve the equation 



8f(x) = 13741, 



or a; 3 -125# 2 + 2500a; -1717 = 0, 



we find that 0"5 substituted for x gives a negative result, while 1 

 gives a positive, which shows that a root lies between 0"5 and 1. 



Hence 8/(8) is greater than the greatest value of /(cj) 

 when 8 has a value somewhere between a half and one mile. 

 Therefore 8 is certainly greater than c x if 8 = 1 mile, which is 

 very much less than the mean depth of the ocean. We may there- 

 fore safely assume 8 — c x as positive for all values of 8 with which 

 we are concerned, and for which the conditions of the problem 

 are fulfilled. 



We say advisedly "for which the conditions of the problem are 

 fulfilled," because our demonstration assumes that the continental 

 and oceanic areas are abruptly separated, whereas in nature they 

 merge into one another, because there is an area of deposition 

 fringing the coasts which is neither oceanic nor continental ; over 

 these areas our formulae will not apply, and indeed there is reason 

 to think that there the assumption of the equality of gravity is not 

 warranted, because in crossing the Atlantic Hecker observed that 

 gravity was slightly in excess as the land was approached; and bhe 



