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IV. u Densities in the Eartlis Crust." 

 By J. Brill, M.A* 



IN the seventeenth chapter of his ' Physics of the Earth's 

 Crust/ the Rev. 0. Fisher has investigated an arrange- 

 ment of the densities and thicknesses of the different layers 

 composing the Earth's crust, which would give, to a high 

 degree of approximation, a uniform value for gravity over the 

 Earth's surface. As attempts have recently been made, in 

 the pages of this Journal and elsewhere, to criticise the method 

 of investigation, it occurred to me that it might be desirable 

 to give an independent investigation of the equations obtained 

 by Mr. Fisher. This might possibly serve to render the 

 meaning of the work clearer than is done in the book. 



In the chapter referred to above the central nucleus is 

 supposed to consist of concentric spherical shells of uniform 

 density, so that it will only be necessary to consider the 

 portion outside this, which we will hereafter refer to as the 

 " crust/' We will suppose this portion to consist of m layers, 

 whose densities, commencing from the outermost, are re- 

 spectively p 1} p 2 , . . ., p m . We will also use the symbols 

 ki, k 2y . . ., k m , to denote the distances of the lower surfaces 

 of the respective layers from the Earth's surface, these dis- 

 tances being measured along a radius of the Earth. This will 

 be more convenient than taking symbols to denote the thick- 

 nesses of the layers. Since we suppose the layers forming 

 the crust to be of varying density and thickness, we see that 

 the p's will vary from one radius to another, as also will the 

 k'sj with the single exception of k m . We must suppose 

 k m constant, as we have laid down that the inner surface of 

 the Earth's crust shall be a sphere concentric with the outer 

 one. 



Let be the Earth's centre, and P a fixed point on its 

 surface. We will take OP as the axis of polar coordinates, 

 and consider the vertical component of the attraction at P of 

 a polar element taken somewhere within the crust. If p be 

 the density of the element we are considering, this will be 

 represented by the expression 



r 2 sind(a — r cos 6) . 1A . , 

 r {r* — 2ar cos 6 + a )* 



o 



a being: the Earth's radius. 



* Communicated by the Author. 



