232 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLID BODIES. 



Here again, we are under the necessity of dispensing with the 

 assistance of a diagram, the investigation being wholly analytical ; in 

 order therefore to proceed, 



Put w = the weight of the body when weighed in water, 

 w'~ the weight when weighed in air, 



s the specific gravity of water, generally expressed by unity, 

 s* nz the specific gravity of air, and 

 x rzz the real weight of the body required. 



Then we have x w' ', and x w> for the weights which the body 

 loses in air and in water ; but we have deduced it as an inference 

 from Proposition V., that when the same body is weighed in different 

 fluids, it loses weights in proportion to the specific gravities of the 

 fluids in which it is weighed ; consequently, we have 



x w' : x w : : s' is; 



therefore, by making the product of the mean terms equal to the 

 product of the extremes, we have 



s (x w') s' (x w), 

 and from this, by separating the terms, and transposing, we get 



(s s') x ~ s w' s'w; 



consequently, by division, we obtain 



__sw' s' w 



s s' (185). 



268. The equation in its present form, supplies us with the follow- 

 ing practical rule for its reduction. 



RULE. Divide the difference between the products of the 

 alternate weights and specific gravities, by the difference of 

 the specific gravities, and the quotient will be the real weight 

 of the body. 



269. EXAMPLE. A certain body when weighed in water and in air, 

 is found to equiponderate 12 and 13.9975 Ibs. respectively; what is 

 its real weight, the specific gravities of air and water being as 1 to 

 .0012? 



Here, by operating as directed in the rule, we have 

 1X13.9975 .0012X12 



x 



.0012 



14 Ibs. 



From which it appears, that a body of 14 Ibs. avoirdupois, will 

 completely fulfil the conditions of the question. 



