.s/A' WILLIAM SIEMENS, F.R.S. 



361 



ount of variation that will be produced in the total attraction 

 of the earth, supposing it to be of uniform density, by a given 

 depth of water below the attracted point P, a line is drawn from 

 that point to the centre of the earth, and the same is divided into 

 an unlimited number of indefinitely thin slices, by planes perpen- 

 dicular to that line. 



In taking one of these slices at the distance h from the attracted 

 point, an expression is obtained representing its aggregate attrac- 

 tion, thus 



The slice is composed of concentric rings of sectional area 

 dh . dx=dh . z . da : cos a, and of the capacity 2ir . z . sin a . dh 



2ir . z . sin a . dh . z . da 

 .z.da: cos a, which gives - - - as the dine- 



3 



rential of the attraction, where z and z in the numerator and z* 

 in the denominator, although variable quantities, always vary 

 together, or ddA^ = 2ir . dh . sin a . da. 



This expression has to be integrated between the limits of h 

 and 0, and a and ; thus 



/* / /"A [a fk 



I 2ir . dh . sin a. da = I 2ndh \ sin a. da = 2ir\ dh (1 - cos a). 

 0/0 J o J o Jo 



/sin a. da=l -COS a, 

 9 



Since 



also 



a = 

 ~ 



/. 27T [ h dh(\ - cos a) = 2* f Yl - -*{ij}dh = 2nh - 27r ^ ~^=, 

 Jo J o \ *j'2ii/ J o J2si 



is the total attractive force exercised by the uppermost portion of 

 the globe to the depth h. 



For small values of /t, the expression A/ -~L may be neglected, 



V 2R 



and the formula may be written 



. . . (2) 



