112 Mr Fisher, Densities of the Earth's Crust 



These being equal, if one of them vanishes, they must all 

 vanish. Hence if t x = t 2 , they must all vanish, because (1) does so. 

 Now t Y = t 2 causes (2) to vanish as well as (1), but none of the rest. 

 Hence ^ = t 2 must necessitate some further relation among the t's 

 which will cause the remaining products to vanish. 



J-I £/j —- v<2, ~~ £3 == ^4 , (\jC.j 



each product will contain a vanishing factor, and they will all 

 vanish. Hence if two layers are equal, the conditions of the 

 problem will be satisfied by their being all equal, which would in 

 effect reduce the shell to a single layer. 



But the conditions might also be satisfied if the t's with odd 

 suffixes are equal to those with even. Thus 



t 1 = t 2 causes (1) and (2) to vanish, 

 t 3 = t 4 causes (3) and (4) to vanish, 

 and so on. 



And if the t's with odd suffixes are in one of the frustra, and 

 those with the even in the other (for making t 2 and t 4 negative 

 will cause no difference in the demonstration), this implies that 

 the layers in the two frustra are of equal thickness. Hence if two 

 layers £ x and t 2 are equal, the conditions will be satisfied if all the 

 corresponding layers in the two frustra are of equal thickness. 

 In this case it is clear that the number of layers in the two frustra 

 must be the same, and in no other way than these two can the 

 conditions be fulfilled, when a layer in one frustrum is equal to 

 one in the other. 



Next consider the products (3) and (4), and divide out the 

 common factor (t 3 — t 4 ). Now make t s = t 4 and we have the two 

 sides of the resulting equation identical except the density factors. 

 These must therefore be equal, or t 3 = t 4 . This shows that two 

 layers in two frustra being of equal thickness, they must also be 

 of equal density. 



Thus we arrive at the remarkable result that if under the 

 conditions of the problem the layers in two frustra are equal in 

 thickness, each to each, they will also be equal in density; and all 

 the other corresponding layers in like manner will be equal each 

 to each in thickness and in density. 



Since in the case of the earth, the upper layer in the land 

 being of rock, differs in density from the upper layer on the side of 

 the ocean, which is water, it follows that all the other layers 

 on the side of the land must differ from those on the side of the 

 ocean in thickness and density. 



We see then that if at an early stage of the world's history it 

 was covered by an ocean of uniform depth, the globe which the 

 ocean covered must have consisted of concentric layers each of 



