48 



NA TURE 



\Nov. 20, 1873 



The advantage possessed by Gauss's method of alter- 

 nation over Borda's method of substitution has been 

 proved by Prof. Miller as follows : — 



Let P and Q be two standard weights of the same 

 denomination to be compared, and C the counterpoise of 

 each. 



For Borda's method, let the readings of the index be 

 denoted by (C, P), when C is in the left pan and P in the 

 right pan, and by (C, O), when C is in the left pan, and 

 O in the right pan. 



For Gauss's method, let (O, P) denote the readings 

 when O is in the left pan and P in the right, and (P, Q), 

 when P is in the left pan and O in the right pan. 



Let (' be the probable difference between the recorded 

 and the true position of equilibrium, that is to say, the 

 probable error of a single luci inking (not of a comparison, 

 which requires two weighings). 



Then by Borda's method, (C, P) has a probable error e, 

 and (C, O) has a probable error c ; and the two weighings 

 give the value of P - (.) with a probable error of 



By Gauss's method, (O, P) has a probable error e, and 

 (P, Q) has a probable trror c ; and the two weighings 

 give the value of P - () with a probable error of 



Thus the probable error of the result of two weighings 

 by Borda's method is twice as great as by Gauss's 

 method. 



To obtain a value of P — O by Borda's method with a 

 probable error of — ^z, we must make four comparisons of 



two weighings each. Therefore one comparison by the 

 method of Gauss gives as good a result as four compari- 

 sons by Borda's method. 



The result of this weighing of two standard weights 

 against each other gives only their apparent difference 

 when weighed in air. In order to ascertain their 

 true difference, it becomes necessary to determine the 

 weight of air displaced by each, from the data which 

 have been already mentioned, and to allow for any differ- 

 ence of weight of air displaced, according to the following 

 formula : — 



If the weights P and Q appear to be equal in air, the 

 weight of P — weight of air displaced by P is equal to 

 the weight of O — weight of air displaced by O. 



In determining the weight of ordinary atmospheric air 

 in rooms where standard weights are compared, and con- 

 taining a certain quantity of aqueous vapour and car- 

 bonic acid, the practice has been to take, as the unit 

 of weight of air, a litre of dry atmospheric air free 

 from carbonic acid, = r2932227 gramme, at 0° C., as 

 determined by Ritter from the observations of M. 

 Regnault in Paris, lat. 48° 50' 14", and 60 metres above 

 the level of the sea, under the barometric pressure of 760 

 millimetres of mercury. Assuming that atmospheric air 

 contains, on an average, carbonic acid equal to 00004 of 

 its volume, and the density of carbonic acid gas being i '5 29 

 of that of atmospheric air, the weight of a litre of dry 

 atmospheric air cont:uning its average amount of car- 

 bonic acid, under the stated circumstances, is i"2934963 

 gramme. 



Allowance should be made for the difference of the 

 force of gravity in latitudes other than Paris, as well as 

 for the difference of height of the place of observation 

 above the mean level of the sea. Although the absolute 

 weight varies with the latitude and with the height above 

 or below the mean level of the sea, yet this variation is not 

 felt in the comparison of standard weights in a vacuum, 

 because the weights arc equally affected on both sides of 

 the beam. But in all weighings of standards in air re- 



quiring special accuracy, such variation must be taken 

 into account in computing the weight of air displaced by 

 each standard weight. 



Mr. Baily has shown from his pendulum experiments * 

 that if we take G to denote the force of gravity at the 

 mean level of the sea in lat. 45", the force of gravity in 

 lat. X, at the mean level of the sea 



= G (i — 0-0025659 cos 2 X). 



And Poisson f has proved that the force of gravity in a 

 given latitude at a place on the surface of the earth at the 

 height z above the mean level of the sea — 



= I , _ ^-1 _ 3 P' \ £_ I V Choree of gravity at the mean 

 ( \ 2p)r) level of the sea in the same lat.) 



where r is the radius of the earth, p its mean density, and 

 p' the density of that part of the earth which is above the 

 mean level of the sea. If as is probable, — 



P :/' = 5 : 11; 



3'' =f32 nearly; /■=6366i98 metres, 



2p 



it follows that the weight in grammes of a litre of dry 

 atmospheric air containing the average amount of car- 

 bonic acid, at 0°, and under the pressure of 760 milli- 

 metres of mercury at o", at the height ^ above the mean 

 level of the sea in lat. X is— 



f293o693 (i - 1-32 — ) (i — 0-0025659003 2 X). 



At Cambridge, where Prof Miller's observations for 

 determining the weight of the new standard pound were 

 made, in lat. 52° 12' 18", about 8 metres above the mean 

 level of the sea (and for which place his tables were com- 

 puted,) the weight of a litre of dry air containing the 

 average quantity of carbonic acid was found by him to be 

 1-293893 gramme. This weight of air is therefore a little 

 greater than at Paris. From similar data, after taking a 

 further correction by Lasch of the weight of a litre of dry 

 air at Paris = 1-293204 gramme, the weight of a litre of 

 dry air at Berlin (lat. 52° 30', and 40 metres above mean 

 sea level) has been computed to be 1-29388 gramme. 



The co-efficient of expansion of air under constant 

 pressure between 0° and 50° C. is taken from Regnault's 

 determination to be 0-003656 for 1° C, in other words 

 between 0° and 50° C., the ratio of the density of air at 

 0° to its density at /° is i + 0003656 /. 



With regard to the barometric pressure of the air and 

 the allowance to be made for the pressure of vapour 

 present in it, the density of the vapour of water is deter- 

 mined to be 0-622 of that of air ; that is to say, the ratio 

 of the density of the vapour of water to that of air is 

 I - 0-378. 



Hence, if / be the temperature of the air, i the baro- 

 metric pressure, 7' the pressure of the vapour present in 

 the air, A and v being expressed in millimetres of mercury 

 at 0° C, the weight of a litre of air at Cambridge becomes 



i'293893 

 I -1- 0-003656 / 



6 — 0-378 V 

 760 



The ratio of the density of air to the maximum density 

 of water is found by dividing the above expression by 

 1,000, as a litre of water is the volume of 1,000 grammes 

 of water at its maximum density. Prof Miller's Table I. 

 gives the logarithms of this ratio at the normal barometric 

 pressure of 760 millimetres, at the several degrees of 

 temperature from o'' to 30". These logarithms require 

 to be diminished only by 0-000026 for weighings at the 

 Standards Office, Westminster, lat. 51° 30', and about 

 5 metres above the mean sea-level ; and when dimi- 



t "M 



of the Astronomical Society," vol. v 

 rinstitut," tome Jtxi. pp. yi. 



1. p. 04. 



3jS. 



