The 18th March, A. M. all having been verified, the comparisons were made as by the following Table:

Standards Microscope Read. Mean of the four Correction for Mi Corrected Result Final Value. Mean Tempera-
compared.

ings.
Results.
crom. Value.
of Reading

ture.
78."8

+ 46.6
Mc+= 0.0440752
M5+1= 0.043000
Changed ends for middle, 0.043900

0.0001756

0.043724 78.756276
Mc+i= 0.043250
M3+1= 0.043235
Mc+j= 0.039175
M5+!=

0.039300
Changed ends for middle, 0.039702 0.0001588

0.039643 -760357 Mc+=

0.040200
M3+?= 0.040135

48.6
Ml+j= 0.041525
M1+1=

0.040830
Changed ends for middle, > 0.041295

0.0001651

0.041130 --758870
0.041600
M7+?=

0.041225
The four double Meter Bars were now successively put under comparison in their four possible positions, and found
A=

0.044475
V =

0.044250
Changed end for end, . } 0.044369 0.0001775 0.0441915 78.7568085
A=

0.044325 V=

0.044425

The same day P. M. the comparisons were repeated as follow :

Mc+= 0.040475 M3+(=

0.040525 Changed ends for middle, > 0.040850

0.0001634

0.0406866 ,7593134
0.040850
Mɔ+=

0.041550
Ml+j= 0.043175
M2+= 0.042450
Changed ends for middle, < 0.0426125 0.0001704

0.0424421 7575579
Ml+=

0.042300
M2+!=
0.042525

51.8
To make these results comparable, it is necessary to reduce them all to ope temperature by the difference of expan-
sion between iron and brass. I shall for this make use of my own results of the pyrometrical experiments made im-
mediately after the comparison, and published in the philosophical transactions of Philadelphia of the same year,
the results of which gave the expansion for one degree of temperature of Fahrenheit's scale,
In iron = 0.0000069635351

Difference.
In brass = 0.00001050903 "

$ 0.000003545495 Decimal parts of the length.
On account of the change of temperature during the comparison, and the considerable influence of this element in
the results, it is necessary to take for each comparison the proportional temperature corresponding to it between the
readings of the thermometer, the work proceeding regularly, each single comparison will correspond to the propor-
tional time.

The standard temperature to which I find it most natural to reduce the measurements is 32° Fahr. or 0° Cen-
tesimal & Reaumur, it being adopted for the meter and toise. I shall reduce the results of the iron to brass, so that
the numbers will express the length of the meters in English inches of the brass scale, both at the temperature of 32°,
which appears to me most naturally, being possible to obtain by actual experiment, while the giving of the meter at
32° in length, on the scale at 62°, is impossible to produce and verify in nature, therefore always a result of calcula-
tion in which the ratio of expansion used has too much influence. The brass metre is therefore considered as need.
ing no reduction.

The constant quantity of 0'000172 is also to be added to each measure taken on account of the individual values
of the part of the scale made use of.

The following table will therefore present the Results of all the foregoing Comparisons, with their Reductions :

Date of compari

Temperature of the

comparison.
31°.4

Result of the com

parison.
78.760718

+

Bars A
49.2

78.7558085 17.2 0.00480286 78.76078336
49.5
-.7568114 17.5

0.00488162 -.76186502 49.8 -.7563810

17.8

0.00497040 -.76152340 50.2 -.7568110

18.2

0.00508204 -.76206509 18th March, Mc+

50.5

78.7553503 18.5 0.00516567 78.76068797 P. M. Mc+i

51.0

-75.93134 19.0 0.00530555 -.76479095
Ml+j
51,5
-.7575579 19.5 0.00544510

-.7631750
By the principles of the arrangement, it is evident that the value of any one single metre compared above will be

(c+b) + (c+1)—(6+1) .
obtained by a simple equation of the form; C=-

; and so any of the others, mutating the letters accordingly.

The final results of these comparisons form therefore the following table of the values of the different meters compared at the temperature of 32° of both metre and scales, in English inches : Date of the com

M.
Mb.

Mj. parison. 15th March,

39.37992415 S9.379854162 18th March, A.M. -.3809641

39.3840606 P. M. -.37955601

-.3836290 Means 39.381022708 39.37972015

39.3838448 Correction of the brass metre by the certificate

. +0.00039381 which applied, gives the metre corrected - - - 39.38024797

These results might now be compared with those obtained by various comparisons made in England, these being however always stated so as taking the metre at 32°, and in value of the English scale at 62°, it is necessary to reduce them all for 30° difference of temperature full expansion of the brass. As I have not now the books in which they are related, and am ignorant, so various are they, which English standard and expansion has been used ; supposing, however, that it has most generally been that of Borda, I will here only present, in a tabular shape, the different results as I have them, and reduce them to 32°, to compare them with my results. Observing, at the same time, that they are yet subject to the differences between the English standards themselves, which are in some instances greater than the differences of these results, as may be seen by the paper of Sir G. Shuckburgh, quoted above, and the account of Mr. Pictet, of Geneva, made in London in 1802. Borda's expansion for brass being 0.00000999, (though I have seen it lately stated at 0.0000101, on what ground I do not know, unless I suppose a mistake.)

Difference with my
Authority or observer.
Value given at Reduced to 32o. results of commit.

tee metre. The Roy. Soc. accepted 39.370572 39.38126801 +0.0002453

- in 1800 -.3702 -.380896 -0.0001267 Mr. Pictet in 1802

.371

381696 +0.0006733 Mr. Kater on S. G.

Shuckburgh's scale, lately, 1818

-.37079 .381486 + 0.0004633 The same on Bird's scale -.37062 -.381316 + 0.0002933 (Unknown wbich of Bird's

scales, there being a difference.)

I do not compare by my ratio of expansion, because they were not made or known at the times of the older comparison ; of course could never have been employed in them.

The 21st March, I took the different standards of the toise under comparison.

The toise of Canivet being half an inch French in thickness, and the brass scale half an inch English, this difference was compensated by laying four thicknesses of white paper strips under the whole length of the scale; the microscopes were adjusted to fit this toise, and then the scale adjusted to it; the other toises had rules of proper thickness to bring them to the same focus.