100 Archdeacon Pratt on Mr. Hopkins's Method of 



Hence the moment of the fluid pressure on this element about 

 the axis of x 



=pr 3 sin O'dfidd 1 (cos I . sin 6' sin </>' — sin I sin <f>' . cos 6) 



=pr 3 sin e'dfidO' sin <£' cos (l-6') = 2epa 3 sin 2 6" cos 6 ! sin ft dd'dfi, 



putting r = a, the mean radius, because the square of e may be 

 neglected. Integrating for the whole surface, putting cos 0' = /u/, 

 the part of L which depends on the fluid pressure 



= 2a 3 e ( * ( 'VVl^-Z^sin <£V#'- 



J -l Jo 



Similarly the part of the moment M which depends on fluid 

 pressure 



= -2a 3 e ( ' Jp/A/r^cos fidfM'dfi. 



The function under the signs of integration is a Laplace's func- 

 tion of the second order. 



5. I must now find p. The centrifugal forces on any particle 

 (#V) of the fluid parallel to x and y are rfix' and rfiy 1 . Also, if 

 R is the distance of the sun from that particle, the attraction of 

 the sun on it 



jL E 

 ~ dR R' 



and similarly of the moon ; and the equation of fluid equilibrium 

 at the epoch gives 



d ^ = jd.? n sm*6' + Sd.^+Md.±, 



r', &, (j> r being the coordinates to the particle of fluid. I shall at 

 present leave out M (the moon), and, when the effect of S is 

 found, add a similar term for M. Now 



1 _ 1 p'V „ r' 2 

 R-c + ^? 



= -+P l -Z+P^ + . 



P v P 2 , . . . being Laplace's coefficients. I shall integrate dp from 

 the earth's centre, along r', to the surface of the crust, keeping 

 6' and <j>' constant. Then 



p = n 2 (| + i - ^ f VrW + | (p x C p 'dr f + 2P q Cp'r'dr' + .). 



Substitute this in the formulae of the last paragraph, observing 

 that as e 2 is to be neglected, the means a and a 1 may be put for 

 r and r l . Observing the properties of Laplace's functions, and 



