ON BALANCES. 69 



A similar mode of noting the weighings is pursued in a 

 comparison by Borda's method, when for the two observa- 

 tions a counterpoise weight remains in the right-hand pan 

 y, and the two bodies A and B are weighed successively 

 against this counterpoise, the difference between A and B 

 being shown in divisions of the index scale by the whole 

 difference between the two resting points of the balance. 



31. It has been mathematically proved that Gauss's 

 method gives twice as accurate a result of a single observa- 

 tion as Borda's method. In fact, as the distance between 

 the two positions of equilibrium, or resting points of the 

 balance, serves to measure the difference between the two 

 weights compared, and this distance is by Gauss's method 

 reduced to one half less than by Borda's method, it is evident 

 that the result of a single observation by Gauss's method is 

 twice as accurate as by Borda's, and therefore, that one 

 comparison by Gauss's method gives as good a result as four 

 comparisons by Borda's method. In other words, the probable 

 error of the result of a comparison by Borda's method is four 

 times as great as by Gauss's method. 



32. The methods of weighings of which I hav\3 spoken 

 give only the apparent difference of weight in air of the two 

 bodies compared. In order to ascertain their true difference 

 in weight it is necessary to allow for the weight of air 

 displaced by each, in other words, to reduce the weighings 

 to a vacuum. 



The formula for this purpose is as follows : 



If the weights A and B appear to be equal when weighed 

 in air, then the weight of A the weight of air displaced by 

 A =r weight of B weight of air displaced by B. 



The justness of this mode of correction is obvious from the 

 following considerations : 



A body weighed in water weighs less than when weighed 

 in air by the difference between the weight of water and 

 weight of air displaced by its volume. In like manner a 

 body weighed in air weighs less than when weighed in a 

 vacuum by the weight of air which its volume displaces. It 

 follows that, for instance, a brass weight, if it weighs exactly 

 one pound in a vacuum, must when weighed in air weigh 

 less than a pound by the weight of air that it displaces, and 

 its true weight, when weighed in air, must consequently be 

 found by deducting the weight of air thus displaced. 



