OF THE EQUILIBRIUM OF FLOATATION. 287 



drawn into the greater and lesser specific gravities of the fluid, 

 and the product will express the required variation in the 

 position of the instrument. 



365. EXAMPLE. Suppose the specific gravity of the fluid to vary 

 from 0.5738 to 0.5926 ounces per cubic inch during the time of the 

 experiment, what is the corresponding variation in the depth of the 

 instrument, its whole weight being 23 ounces, and the diameter of the 

 upright stem equal to one twelfth of an inch ? 



Here, by attending to the directions in the rule, we obtain 

 . ., 23(0.5926 0.5738) 



Hence it appears, that by a difference of 0.0325 in the absolute 

 specific gravity of the fluid, there arises a difference of 233 inches in 

 the position of the instrument ; this seems a very great difference, and 

 is in reality far beyond the bounds prescribed for the whole apparatus 

 to occupy ; it serves, however, to exemplify the extreme delicacy of 

 the principle, and when the changes in the specific gravity are very 

 minute, the corresponding changes in depth will nevertheless be suffi- 

 ciently distinct to admit of an accurate measurement. 



366. By diminishing the diameter of the upright stem, or increasing 

 the entire weight of the instrument, which is equivalent to an increase 

 in the weight of the fluid displaced, the sensibility of the aerometer 

 may be greatly increased. This is manifest, for by inference 5, 

 equation (202), it will readily appear, that if the specific gravity 

 remains the same, the quantity by which the instrument sinks in the 

 fluid on the addition of a small weight w r , varies directly as the mag- 

 nitude of the weight added, and inversely as the square of the radius 

 of the upright stem. 



Let us suppose, that by the addition of the small weight w', the 

 length of the part of the stem Z, which is originally immersed, becomes 

 equal to /' ; then, by the principles of mensuration, the increased 

 magnitude of the immersed stem is 7rr*(l' I); but the weight of a 

 body is equal to its magnitude multiplied by its specific gravity; 

 hence we have 



*r\l' l)s = w'; 



and this, by division, becomes 



r-r=+-. 



irr^s 



Now it is obvious, that by the supposition of a constant specific 

 gravity, the quantity ITS is also constant : it therefore follows, that 



