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of K were caused by this bit of wood being weighed with it after the first comparison of K by Captain Kater, and by the gradual oxidation of the surface of K. The discrepancies presented by the weighings of the brass troy pounds at different times, due to the effect of oxidation or other causes, are so large, that I resolved, with the consent of the Astronomer Royal, to rest for the evidence of the weight of the lost standard entirely on the 300 comparisons of Sp and the 140 comparisons of RS with U.
If we consider the discordances presented by the weighings of the brass troy pounds simply as errors of observation, without paying any regard to their probable causes, the resulting value of U will not be very different from that given by the platinum troy pounds alone. By the observations of 1824 and 1829,
By the observations of 1844,
=Sp +0.0057 Sb
=Sp +0.0030 K
+0.0363 Ex +L+Ed +D=2(Sb + K) +0:0260
Whence, supposing the errors of weighing in 1844 to be insensible, compared with the discordances of the brass troy pounds,
weight. (1) U=Sp+0.0081
30 (2) U=Sp+0.0087
14 (3) U=Sp+0:0133
6 (4) U=Sp+0.0024
9 (5) U=Sp+0.0261
The mean of all the equations gives U=Sp+0.0096 grain.
Excluding the last, which depends upon the weighings in 1824, U=Sp +0.0079 grain.
Excluding all except the results of the comparisons of U with the two platinum troy pounds, U=Sp+0.0083 grain.
The temperatures were determined by means of three thermometers by Bunten, having centesimal scales etched upon the tube, and two thermometers having arbitrary scales traced upon the tubes with a diamond point. The zero-points of these were determined at distant intervals. They were often compared with each other, and, lastly, with an excellent standard thermometer constructed at Kew under the directions of Mr. Welsh, in order to form tables of the errors at any point of their scales, and to determine the position of their zeros at any given time. The barometer employed was a portable cistern barometer by Ernst. of Paris, the scale of which was divided into millimetres. It was compared first with the standard barometer of the Paris Observatory, and afterwards with a standard barometer, having a tube of very large bore, belonging to the Taylor Library of Sidney Sussex College, Cambridge.
According to Ritter (Mémoires de la Société de Physique de Genève, t. iii. p. 361), the observations of Regnault show that in Paris, lat. 48° 50' 14", 60 metres above the mean level of the sea, a litre of dry atmospheric air, containing the average amount, 0.0004 of its volume, of carbonic acid, the density of which is 1.529 of that of atmospheric air at 0° Cent., under the pressure of 760 mm. of mercury at 0° Cent., weighs 1.2934963 gramme. If G be taken to denote the force of gravity at the mean level of the sea in lat. 45°, the force of gravity in lat. 1, at the mean level of the sea, =G(1-0.0025659 cos 21) (Baily, Mem. Ast. Soc. vol. vii. p. 94). The force of gravity in a given latitude at a place on the surface of the earth at the height = above the mean level of the sea
x force of gravity at the level of the sea in the
2 same latitude, 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 (Poisson, Traité de Mécanique, t. ii. p. 629).
According to Regnault, the expansion of air under constant pressure from 0° to 100 Cent., is 0:36706 of its volume at 0° Cent.; also at 50° Cent., the mercurial thermometer is about 0°•2 in advance of the air thermometer (Mémoires de l'Institut, t. xxi. p. 91. p. 238, Annales de Chimie, 3 série, t. v. p. 99). Hence, density air at
0°: density air at t=1+0.003656t. The density of the vapour of water is 0.622 of that of air. Hence, if t be the temperature of the air in centesimal degrees, 6 its barometric pressure, v the pressure of vapour, both in millimetres of mercury at 0° Cent., the weight in grammes of a litre of air at a place on the surface of the earth at a height z above the mean level of the sea in lat. 1, will be
(1-0.0025659 cos 22).
Regnault finds that in rooms not heated artificially, the pressure of vapour is two-thirds of the maximum pressure corresponding to the temperature (Memorie della Società Italiana della Scienze in Modena, t. xxv. p. 1).
The weight of air used in reducing the weighings was calculated from the above expression.
The mean rate of expansion of brass, for 1° Cent., from O' Cent. to 100° Cent., usually assumed 0.0000187 of its length at 0° Cent., is considerably larger than the rate of expansion at ordinary atmospheric temperatures, according to the observations of Mr. Sheepshanks, who found that at about 17° Cent. the coefficient of the linear expansion of brass =0.00001722 for 1° Cent. This value of the expansion has been accordingly adopted.
The linear expansion of platinum is assumed to be 0.00000900 for lo Cent., as given by Schumacher in his first table (Phil. Trans. 1836). The expansion of water is calculated from a mean of the experiments of Despretz, I. Pierre and Kopp, corrected for the error of the assumed expansion of mercury by Regnault's observations, and assuming the temperature of maximum density to be 39.945 Cent., in accordance with the result obtained by Messrs. Playfair and Joule. The logarithms of the expansion to 7 places considered as integers, are given with sufficient accuracy, between 4o Cent, and 25° Cent., by 32:72(t-3.945) -0.215(t-3.945)".
Though it appears that only two of the nine weights with which U was compared in 1826 and 1829 are in a state of unexceptionable preservation, and that the number of trustworthy comparisons is reduced from 669 to 440, these are amply sufficient for the purpose of ascertaining the weight of U in air (t=65°.66 Fahr., b=29-75 inches). But in order to find the absolute weight of U, or indeed
its apparent weight in air of a density different from that which it has when t=65°:66 Fahr., b=29.75 inches, a knowledge of the volume of the lost standard is requisite. An indirect way of arriving at it was suggested by Professor Schumacher, by an examination of certain Parliamentary Reports, presented May 26, 1758, April 11, 1759, March 2, 1824. It appears from the first of these, that Mr. Harris, then Assay Master of the Mint, presented to the Committee three troy pounds made under his direction, one of which was the lost Imperial standard ; and from the third, that one of the two remaining pounds came into the possession of Mr. Vandome, and the second into the possession of Mr. Bingley of the Mint. Professor Schumacher then observes that we can still either determine, with the highest degree of probability, the density of the lost Imperial standard, or know with certainty that all hope to arrive at this knowledge is lost. It will be only requisite to ascertain with the greatest care the densities of both pounds, the one in the possession of Mr. Bingley, the other in the possession of Mr. Vandome. If the density of both is found the same, we might from that circumstance draw the highly probable conclusion, that the three single pounds of Mr. Harris, according to my hypothesis, were really made of the same identical metal; and the density of the two remaining pounds might with safety be considered as that of the lost standard. If, on the contrary, the two remaining pounds prove to be of different densities, the hypothesis that all three were made of the same metal is evidently erroneous; and nothing can be inferred from the density of either of the two remaining.
Mr. Vandome readily consented to allow his troy pound to be experimented upon by the Committee. Denoting this weight by the letter V, by weighing in air and in water it was found that AV=8-15084, and that it was about 0.309 grain lighter than U.
Mr. Bingley had in his possession two troy pounds, both dated 1758. One of these, 0, said to be the original weight from which the standard was made for the House of Commons in 1758, has since been purchased by the Committee; the other, M, has been presented to the Mint by Mr. Bingley. As Mr. Bingley was unwilling to permit either of these weights to be weighed in water, Messrs. Troughton and Simms were commissioned to construct an instrument on the principle of the Stereometer invented by M. Say for the purpose of finding the density of gunpowder (Ann. de Chimie, 1797, t. xxiii. p. 1), but with some improvements which I had described in the Philosophical Magazine for July and December, 1834, vol. v. p. 203. Let v prefixed to the symbol of any weight denote the volume of that weight at 0° Cent., the unit of volume being the volume of a grain of water at its maximum density. Then, by means of the Stereometer, it was found that vV — v(= 22.68, OV - OM=17:38. These differences show that the volume of lost standard cannot be inferred with any high degree of probability from a comparison of the remaining pounds. The only resource now remaining was indicated by Professor Schumacher's remarks on the figure of the lost standard :-“As soon as the Imperial standard troy pound was brought to Somerset House, Captain Nehus's first care was to make an accurate drawing of its shape and marks, measuring all its dimensions with the greatest care. The annexed drawing represents this pound in its actual dimensions; and is now, since the original has been destroyed by the calamitous fire that consumed the two Houses of Parliament in 1834, the only thing remaining which can preserve an idea of it.” By a comparison of the figure of U in the Philosophical Transactions for 1836, with a profile of V traced mechanically, the axis and the extreme diameter of the knob and cylindrical portion of U, appeared to be a very little greater than the corresponding dimensions of V. On comparing the profiles of U and V, it did not seem possible to suppose that the volume of U was less than that of V. But the volume of O, as well as that of M, being less than that of V, it appeared that of the three weights V, O, M, V approximated most nearly to U in volume. As the existing data were utterly insufficient to determine how much, if at all, U exceeded V in volume, it appeared safest to assume the volumes of U and V to have been equal. This course was also recommended by Professor Schumacher.
It was afterwards found that 0 was 0.144 grain lighter than U, 40=8.4004; and that Mwas 0.047 grain lighter than U, AM=8:3491.
In a letter from William Miller, Esq., of the Bank of England, dated August 22, 1855, I was apprised of the existence of a fourth troy pound of 1758. This weight was 0.249 grain heavier than U; its density = 8.3175.