378 Rev. 0. Fisher on 



If the attractions are the same in the two cases for any length 

 whatever of the sector, i. e. for any number of different 

 assumed values of 0, the coefficients of the several functions 

 of the variable must be separately zero. Therefore 



&c. = &c. J 



> (A) 



Hence what Mr. Blake says appears to be true, that if the 

 vertical attraction at the apex of a cap sector in the case of 

 any assumed law of attraction can be expanded in the above 

 form, the same proposition regarding the thicknesses and 

 densities will hold good. This will be a remarkable 

 property of the sphere, but it will by no means invalidate any 

 conclusion we can draw from it in the case of the Newtonian 

 law. 



Mr. Blake's second objection is that "the same form of 

 equation would result if we had expressed the same supposed 

 arrangement of layers differently, e. g. if we had taken them 

 as non-overlapping, or if we had divided one into two, each 

 of half the thickness." This remark shows that my critic 

 has not appreciated the idea correctly. The quantity a in the 

 expression is the radius to the outside of each layer, and this 

 being the same for them all, they must overlap. 



The third objection is, that " it is necessary to assume the 

 equality of the two sides to the same degree of approximation 

 as there are layers in the crust." Not at all! I have 

 deduced from the equality that two equal and similar areas 

 (or " patches ") of any form, of which the layers, whether 

 the same in number or not, and densities are so related, will 

 produce the same gravitational effect at any place, one as the 

 other. Now since gravity is known to be the same all over 

 the ocean, we must have 'the gravitational effect of an area of 

 flat land the same as that of an oceanic area of the same size 

 and form, and consequently the layers in these two areas 

 must be related, as shown by the equations (A), no hypothesis 

 as to equality in the numbers of layers being made. For 

 example, let there be an area of land at P, and a similar one of 

 ocean at Q. Take a point R in the ocean equidistant from P 

 and Q ; then, remembering that the layers are understood 

 to be underlaid by a centrobaric nucleus, the contribution of P 



