353 Archdeacon Pratt on the Thickness of 



fluid against a portion of the crust of which C B C is the inner 

 section when the moon is at M, above what it is when the moon 

 is in the prolongation of the radius E F at right angles to E A. 

 The earth and moon are supposed at rest. 



E the earth's mass ; M the moon's, = y'jE ; 



EA = fl', 4000 miles; AB = ^, CD = ^',EB = 6, c= the distance 

 of the moon from E = 60<2. 



If p be the density of any point of the fluid ; r, s the distances 

 of that point from E and the moon, then 



dp E , M , 



-^ = zrdr g ds, 



p er s^ 



by the laws of fluid equilibriurn. Now p is variable ; but at 



all points along the inner surface of the crust it is constant, and 



= J) suppose. Hence integrating along the surface of the crust, 



putting ?- = i, and taking s to apply to any point P in the surface, 



we have 



, , DE , DM , -^,^/l 1\ 



p= constant — jr-or^H =A+DM( J, 



2«"^ s \s c) 



where A is a constant independent of the moon's action. Let 

 6 = angle PEB ; then s= c— h cos Q, neglecting very small quan- 

 tities. Hence 



» = A + DM — COS0. 

 c 



15. I will now determine the resultant force which the part of 

 this pressure which depends upon the moon produces on the 

 part of the shell CBC in the direction EA. 



The ax'ea of an elementary ring of the surface =27rb sin 6 . bdO. 

 The pressure on this, resolved in the direction EA, 



0_.7,3 



= l!JLDMsin^cos2^rf6'. 

 c^ 



Hence the resultant pressure in EA 



= ^^DM(l-cos3«). 

 o c 



16. The ordinary pressure of the internal fluid sustains the 

 proper proportion of the crust, viz. cCBC'c'Ac. The additional 

 part, defined by the revolution of the triangle CDc around EA, 

 by its weight counteracts in part the effect of the fluid pressure 

 caused by the moon, and must be taken into calculation. 



Let A be the area of the triangle CDc, and h the distance of 

 its centre of gravity from EA ; then A . h is the volume of this 

 mass. Its several parts press down towards E, and its average 

 density is less than D. I shall therefore not be under-estimating 



