166 W. P. Trowbridge on Deep Sea Soundings. 
therefore consider the force of gravity constant; and in realit 
its diminution due to depth will be less than its variation wit 
a the latitude on the surface. 
ge The density of the water increases with the depth, according 
; to the law of compressibility of water, and to find the variations 
of density with depth we must refer to the law of compressi- 
bility. 
From a report of Mr. Schott it appears that the shortening of 
a column of water from the pressure due to 600 fathoms is two 
fathoms; or the cubical diminution is about -0001 of volume 
for one atm ere. é 
The density being inversely as the volumes, for a give 
weight, we shall have, supposing (D) to be the density at the 
surface, and (D’) the density at the depth (z) . 
D'=D. 
z 
1i—-0001. 33°90 
From this expression we find the density at the surface equal 
1-029, at 1000 fathoms 1:048, at 2000 fathoms 1-066, 3000 fathoms 
1-084. This increase of density is too small to affect sensibly 
the buoyancy of the fluid, or the resistance to motion, when 
considered with reference to the quantities which we wish to 
determine. 
The weight of a 32-pound shot, for instance, in water, at the 
surface will be 
W=32—32 — 
the numerator and denominator of the last term being the spe- 
ific gravities, respectively, of water and cast iron. | 
At 3000 fathoms the weight of the shot would b : 
1-084 4 
72° | 
and we see that the variations fall within the range of the spe- 
cific gravities of the different varieties of cast iron. 
The variation as affecting the resistance to motion will be 
shown in the expression 
W=32— 
R=S.D (m+ q)h, 
where it is impossible to determine (m+) or h within the range 
____ Of variations of D. We may therefore regard all the quantities 
a8 constant, and known, in the expression 
. W—(S.D(m-+-9).h.)—R'=0, s 
f which represents the resistance upon the line, and 
etermined from the equation. 
ic gravity of the line be very different from that 
