" Densities in the Earth's Crust,'" 377 



In fact, „ m . 6 



ft*) -1 + smy 



#W — (l+jdnj+l) 



4 sin n 

 2 



^ w== 3 + 8 sm 2 + — rr 



24 sin 3 

 2 



&c. = &c. 



This answers Mr. Blake's objection that " the definite point 

 where the [supposed] fallacy comes in is the assumption that 

 /(#), 4>(6), ^r(O), &c. are independent." They are not inde- 

 pendent, but are all functions of the same variable. 



I will now prove my proposition in a slightly different 

 form from that given in my book, out of regard to Mr. Blake's 

 remark that the method is independent of the law of gravi- 

 tation. 



If we are at liberty to assume any other law than the New- 

 tonian, it will be necessary to introduce a factor C to make 

 the dimensions right. Then the vertical attraction of a cap 

 sector, of density r and thickness t, may with this addition 

 be expressed in the above form. Let this be overlapped by 

 another cap sector, of thickness t 2 and density t 2 , or by any 

 number m of cap sectors. Then the vertical attraction of the 

 composite cap sector at A will be 



C { % m {rt)f{ff) +2 m (T t ^<j>(6)+t m (r^t (9)+ &c. }. 



Now suppose this composite cap sector to be replaced 

 by another of the same areal dimensions with n layers of 

 thicknesses t' &c. and densities t 1 &c. The vertical attraction 

 of this other cap sector at A would be 



Ca { 2„(tV)/(0)+2„(t'^(0) + 2 (t'5)*(0) + &c.}, 



and the difference between the vertical attractions which the 

 two sets of layers would produce would be 



C«{ (2 m {Tt)-Zn(T't'))f(0) +(2«(T^)-2n(Tr a ))^ + &C. }, 



