674 
DR. C. W. SIEMENS ON DETERMINING THE DEPTH OE THE SEA 
also 
h h h h hi 
COS + {( 2 E_A)^ + 42 }i— V2R h~W? 
••• 2rf <B (l-cos«)=2^J j , ‘ (l-X|) A=2A-2rf A 
=2^-2 * ^5 • I • A»=2xA (l-f ^/X) =A, (1) 
is the total attractive force exercised by the uppermost portion of the globe to the 
depth h. 
For small values of h, the expression /y/~ may be neglected, and the formula may 
be written 
A x =2 *h (2) 
In substituting 2R for h in formula (1) we obtain 
A— f R . x, 
the expression for the total attraction of the earth, which was determined by Newton ; 
a verification is thus furnished of the correctness of the above calculation. 
The proportion between the attraction exercised by the upper segment and the whole 
earth, supposing them to be composed of uniform material, is therefore as 
Aj : A=2 nil : -f Rx 
or as 
h : -f R. 
Ratio of variation of Attraction, as the depth to the Earth’s radius. — If sea-water had 
no weight, the total force of gravitation at the point P would be diminished in the ratio 
depth of sea 
| radius ’ 
but, inasmuch as the ratio of the difference of mean rock and sea-water to mean rock is 
2-763-1-026 1*737 
2-763 — 2-763’ 
it follows that the real influence of depth, on the supposition of the earth’s density 
being throughout that of mean rock, would be represented by the expression 
ipih * 
2-763 h h 
|B 614 106 R’ 
579 
or approximately as the depth to R. 
Thus, for a depth of one thousand fathoms, gravitation diminishes by 3 /9 f of itself. 
Necessity for modifying result, neither compression great enough to he sensible in its 
