114 Mr Fisher, Densities of the Earth's Crust 



Accordingly the equation 



t m {rt) = ^ p {r't') 

 becomes 

 O - p 1 )S + ( Pl - p 2 ) (8 + h) + (p 2 - a') (8 + k, + k 2 ) + a'd 



= (fa - fa) d + (fa - cr) (Cj + c 2 ) + crd. 

 And transposing, 



O - p x )8 + (p! - p 2 )(8 + h) + (p2 ~ o-') (8 + h + k 2 ) 



+ (a - o-) d + (fa - (p,) d + O - fa) (d + c 2 ) = 0. 



The equations being now carried to the fifth power of t, terms 

 of the order of (t/a) 41 are retained, which will give a close approxi- 

 mation. 



Comparing the above equation with the general formulae 



Tjtj + T 2 t 2 + T 3 t 3 + &c. = 0, 



&c, &c, 



the equations 



Tltfi (^i - 1 2 ) (fa - t z ) (ti - Q = r 2 t 2 (t 2 - ti) (t 2 - t s ) (t 2 - Q, 

 &c. = &c, 



by substituting the proper values for the t's and t's, give the five 

 equations 



( /ji -p 1 )8(-k 1 )(-k 1 -k 2 )(8-d)(8-c 1 )(8-c 1 -c 2 ) (1) 



= ( Pl - p 2 ) (8 + h)h (- k 2 ) (8 + h- d) 



(8 + h- Cl ) (8 + h- d- c 2 )...(2) 



= (p 2 -<r')(8 + k l + k 2 )(k 1 + k 2 )k 2 (8 + k 1 + k 2 -d) 



(8 + ^ + k 2 - d) (8 + k x + k 2 - d - c 2 ). ..(3) 



= (a'-<x)d(d-8)(d-8-k 1 )(d-8-k 1 -k 2 ) 



(d-cj (c?-d-c 2 )...(4) 



= (fa - fa) d(ci - S)(d -8- A? x )(d -8-k x - Ar 2 )(d - d)(- c 2 ) . . .(5) 



= (o- - </> 2 ) (Cj + c 2 ) (Cj + c 2 - 8) (d + c 2 - 8 - &j) 



(d + d - S - & x - k 2 ) (d + d - d) c 2 . . .(6). 



Of the thicknesses here involved, d exceeds all the others on 

 either side. Of the rest it is easy to see which factors are positive 

 and which negative when the thicknesses are on the same side; 

 but when they are on different sides, as for instance 8 and c 1} the 

 signs of the factors are not self-evident. The density factor (p,—p x ) 

 is negative, because the density of water is less than that of any 

 kind of rock; and we may safely assume that the density of the 

 upper layer of the land consisting chiefly of sedimentary rocks, 



