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Phil. Mag. S. 5. Vol. 39. No. 241. June 1895.

2 L

the actual temperature on the French hard-glass mercury scale as determined by the Tonnelot. The second column gives the corresponding readings of "Joule A." As Joule, in his work, assumed a fixed zero of his thermometer, we must reduce the observations here also in the same way.

Any convenient position may be assumed as zero, as the scale-value which is to be deduced from the observations will only depend on the differences of readings, so that the zero is really eliminated. But it is convenient to take as zero that corresponding to the average temperature of the air, which in our case was about 23:33. The third column gives, therefore, the numbers obtained by subtracting 23.33 from the readings given in the second column. If Joule's scale-value is correct these figures should, when multiplied by his factor, give the temperature as determined by a thermometer made of glass having the composition of these thermometers. Joule's reducing factor is 0.077214, which for the Centigrade scale becomes 042897. For convenience of calculation I have taken it as '0429. The fourth column gives the numbers so reduced. Columns V. and VI. give the corrections to the vertical position and the corrected readings. The last column gives the differences between the temperatures as determined by the Tonnelot and Joule's thermometer respectively. These numbers show no very marked increase or diminution between the temperatures of 10° and 30°. If the numbers in column VI. were constant throughout, it would mean that the two thermometers read alike as regards differences of temperature.

In order to obtain the greatest possible information from the numbers obtained they were reduced by the method of least squares, all comparisons below 13° and above 22° being left out of account as lying outside the range within which Joule worked. If T; represents the reading on the Joule thermometer, Tr that on the Tonnelot, and we wish to form an equation

Tr−T;=a+bTT

',

we may do so, substituting for TT-T; the number in column VI., and for Tr those in column I. The constants a and b were thus found to be

a=0·0081, b=0·000933±·00068.

If we denote by t and t; intervals on the two thermometers we finally find

t;=tr(1–000093).

The result of this calculation, therefore, would be that the scale of the Joule thermometer is about one part in a thousand smaller than that of the Tonnelot, the difference being due either to a difference in the glass or to faulty calibration.

In order to compare the observed differences in the readings of the two thermometers with the values calculated from the most probable scale-value of the Joule, I have added columns VIII. and IX., the former giving the calculated value of TT-T, and the latter the difference & which is either due to errors of observation or to irregular errors of graduation of one or other of the thermometers. Although the obvious fault in this respect shown by the Joule prepares us for occasional differences of about 0°-01, I was not, for several reasons, satisfied with the results of this series of comparisons. The apparatus had not reached its final form during these experiments, the stirring was not as good, and the thermometer had not yet been protected against the inflow of cold water through the opening in the roof of the inner box. A great difficulty was also found in comparing together directly the Joule thermometer, which was rather sluggish in its motion, with the Tonnelot, which answered very quickly the smallest change of temperature. Unless care was taken, therefore, to make the rise exceedingly uniform errors were easily made. Additional uncertainty was introduced by the frequent redeterminations of the zero of the Tonnelot. The probable error of the calculated coefficient was too great to allow me to be satisfied with its value.

A second series of experiments was therefore decided upon, and as in a joint research in the equivalent of heat I had occasion, together with Mr. Gannon, to determine with considerable accuracy the scale-value of a Baudin thermometer graduated directly to a 50th of a degree, I made use of the latter in the second series.

The experiments were made exactly in the same way as before, the Joule being directly compared with the Baudin, and zero readings being dispensed with. The results are embodied in Table VI. The first column gives the temperatures according to the Joule thermometer, the coefficient 0429 being again used, and the readings being converted to the vertical position. The second column gives the reading according to the Baudin thermometer, after the proper calibration correction had been applied and the reading also. reduced to the vertical position. The third column gives the difference between the numbers in the two first.

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The figures of this table were reduced in the same way as those of the first series of measurements.

TB-T; = a+bТB

we find by the method of least squares

a=-'0017,

b=+00092+00022.

If we write

The values of TB-T, calculated by this formula are entered into the fourth column of Table VI. The differences & between the calculated and observed values, which are also given, are seen to be as small as can be expected, never rising to more than 0.006. This series having yielded a satisfactory comparison, we must reduce the scale-values obtained by applying the scale-correction of the Baudin thermometer. Denoting the intervals as read off by the thermometers by the small letter t, the above reductions give

t;= t(1-00092).

A small correction is necessitated by the fact that a slight

error was discovered in the pressure-coefficient of the Baudin thermometer after all the above reductions had been made. The corrected interval equation becomes

t; = tÅ(1—·00084).

The comparison between the Baudin and Tonnelot thermometers made by Mr. Gannon and myself had given

ttp-00089 tp.

Hence, by combining the last two equations, t; = t1(1+·00005).

This comparison would therefore show that the Joule and Tonnelot thermometers read exactly alike.

In all these measurements the Baudin and Joule were always read like calorimeter thermometers, without regard to the change of the freezing-point, while the Tonnelot was referred in every case to its proper zero. The equality of the scalevalue of the two thermometers does not hold when they are both read in the same way, but the same interval read on the Tonnelot would be about one part in a thousand smaller than if read on A.

We may combine the results of the two series of comparisons by giving each weights inversely proportional to the probable error of the quantity denoted by b.

We therefore find as the most probable value for

t; = tÅ(1—·00027).

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Without attaching undue importance to this number, we may say that it represents the relation between the Tonnelot standard and Joule's thermometer as accurately as the divisions and calibration of the latter will allow us to judge. The number seems certainly not to be in error by more than one part in a thousand, and probably by less.

The transition to the nitrogen and hydrogen scale may now be made. Using Chappuis' experimental investigation on the French hard-glass thermometers, it is found that a temperature of 16°.5, to which Joule's last equivalent determination refers the interval on the Tonnelot thermometer, is to be diminished by 00268 or 00305*, according as we want to obtain the interval on the nitrogen or hydrogen scale. Thus writing t=t(1-00268),

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*See Schuster and Gannon, Proc. Roy. Soc. lvi. p. 28

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