OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 243 



and in like manner, the weight lost by the cone cab, is found to be 



w"m"s. 



But the weights which the several bodies possess in the fluid, are 

 manifestly equal to the difference between the absolute weights and 

 the weights lost ; and the absolute weights are equal to the magni- 

 tudes drawn into the specific gravities ; therefore, we have 



ms' ms zz m (s' 5) zz the weight of the cone in the fluid, 



3ms' 3ms3m(s' s)zzthe weight of the cylinder, 



m"s' w" szzm" (s r s) zz the weight of the remaining cone. 



Now, if from the weight which the cylinder possesses in the fluid, 

 we subtract the corresponding weight of the cone, and to the remainder 

 add the weight of another cone of equal magnitude and half the 

 specific gravity ; then, the reduced weight of the cylinder in the fluid 

 becomes 



2ws' 4- J*w*' 3mszzwi(2Js' 3s). 



But according to the conditions of the problem, this weight is to 

 be in equilibrio with the weight of the remaining cone; therefore, by 

 the property of the lever, we have 



ro(2js' 3s) : m"(s' s) : : 3 : 1 ; 



and from this, by equating the products of the extremes and means, 

 we get 



3m" (s' s)=m (2 Is' 3s), 



in which equation ra" is unknown ; in order ^therefore to determine its 

 value, divide both sides of the equation by 3(s' 5), and it becomes 



6(s' s) ' (190), 



But this that we have determined, is the magnitude of the part 

 which remains, whereas the problem requires the magnitude of the 

 part to be cut off; now, the magnitude of the whole cone is m ; con- 

 sequently, by subtraction, we have 



__ m(5s' 65) ms' 



- 6 (s--s) : - 6(7=0' 09')- 



289. The practical rule for reducing this equation is very simple, 

 it may be expressed in the following manner. 



RULE. Multiply the magnitude of the cone by its specific 

 gravity, and divide the product by six times the difference 

 between the specific gravity of the cone and cylinder, and that 

 of the fluid, and the quotient will give the magnitude of the 

 part to be cut off, in order to restore the equilibrium^ 



R 2 



