188 SEE— FURTHER RESEARCHES ON 



[April 24, 



the pressure throughout the earth's mass is given by the formula 

 (cf. A. N., 4104) 



where r is the radius of the earth, g mean gravity, q the constant 

 for Laplace's law, 2.52896 radians ^144° 53' 55". 2, o- the density 

 at any point, 8 the density at the surface, and a^ the mean density. 

 To render this expression available for integration throughout 

 the sphere occupied by the earth's mass, we must put for o-^ its value 



., „ sin" {qx) 



and for 8- its value 



^., . sin^ q 



^- = ^'- f ■ 



corresponding to the surface where .r= i. Thus we obtain 

 - _ 3K.r)'^' r sin^(^ ^') s[n^^-| 



For the total pressure throughout a sphere of radius p = rx, r being 

 the external radius, and x= (p/r) = fraction of the radius, we have 



P = \ p ■ ^7rr'x~ • rdx 



^" (3) 



which by integration becomes 



„ 3(o-o^)V47r;'^/^7-^"- sin(^.i-)cos(^,r) . , .v'\ 



As our integration is to include the whole sphere of the earth, we 

 put X =■ I, and then we have 



?■ ■ r^TTf^ I q — sin q cos q sin' q\ 



\g)q~ \ 2q Yl' ^^■ 



p _ 3 K<^)^ ■ ^'4'^^'^ / ^7 — sin ^ cos ^7 sin' ^ ' 

 ~~ 2(0-, 



The total volume of the earth is (4/3)77;-^, and hence the average 

 pressure per unit of area on all concentric spherical surfaces is 



