'of the Earth's Crust. 423 



" 1. That for values of a different from b, the ordinary rules 

 of diflferentiation strictly apply. 



"2. That for the value a = Z', the same rules hold, and that 

 the diflfcrential equation arrived at will apply to either of the 

 regions denoted by « < Z*, « > Z», according as we substitute for 

 y, u,v the values /i(ff), 0,(«), -^lia) ovf^{a), (j)z{a), '^<2{a)- The 

 continuity of the functions across the bounding surface is not 

 therefore assumed in the application of the ordinary rules of the 

 differential calculus. 



" It seems to me that the use of the term ' discontinuous ' 

 function may lead to some misconception. The more correct 

 account of the matter seems to be this : — The functions which 

 represent the density and ellipticity of a stratum from a = to 

 a = b, although mathematically continuous for greater values^ 

 lose at that point their physical meaning; and similarly the 

 functions which represent the density and ellipticity of an exte- 

 rior stratum, although mathematically continuous for values of 

 a less than b, do not acquire a physical meaning until that 

 value is reached." 



The foregoing is sufficient, I believe, to defend me from the 

 charge of bad logic brought against me by Archdeacon Pratt. I 

 now gladly dismiss the personal question, and turn to the consi- 

 deration of the thickness of the earth's crust. 



I shall consider the subject under the three points of view in 

 which Mr. Hopkins, Prof. Hennessy, and I myself have ex- 

 amined it. 



First. Mr. Hopkins has made two suppositions : — 



A. That Laplace's law of density is the law of nature ( —\ 



this law holding good for both the crust and nucleus. 



J8. Having assumed this law, and calculated from it the 

 amount of lunisolar precession, Mr. Hopkins assumes the small 

 difference between the result of calculation and observation to 

 be a 7'eal difference. 



f' The logical inference from these two hypotheses would appear 

 to be that one or other of them is erroneous : Mr. Hopkins 

 thinks differently, and reconciles Laplace's theory with observa- 

 tion by introducing a fluid nucleus without friction, and so cal- 

 culates a thickness for the crust of the earth. I am most 

 anxious to do every justice to the ingenuity and skill with which 

 Mr. Hopkins has worked out his problem, but I must be allowed 

 to doubt whether it is of any geological value. In the first place, 

 I do not believe in Laplace's law of density; and in the second 

 place, even admitting this law, the omission of friction, jmd 

 assumption of perfect fluidity in tlic molten nucleus, nmst be 

 regarded as so far from the truth as to reader the thickness 



