errors of the 32° line. Comparisons were made between the standard 230 and the A and B thermometers and between 230 and 21472 in January, 1876, at 59°.5 F. This work gives us an indirect comparison between the A and B thermometers and 21472. The results are as follows:

At 590.5, the stems of the thermometers being horizontal, the mean of the readings of A1, A2, A3, A, having had Colonel Clarke's corrections, and also the corrections for subsequent change of freezing-point applied to it, and 21472 having had the Kew correction and also the correction for subsequent change of freezing-point applied to it, 21472 then read 0°.06 F. greater than the mean of A1, A2, A3, A4.

Similarly, 21472 corrected reads, 0°.02 F. greater than the corrected mean of B1, B3, B4, at 59°.5.

B2,

As the Kew corrections are only given to the nearest tenth of a degree, this agreement, taken in connection with Prof. Sainte Claire Deville's comparisons with another standard which gave no greater discrepancies, indicates that Casella No. 21472 is very accurately constructed, and that, so far as a mercurial thermometer is concerned, the probable error of its corrected indications is but a few hundredths of a degree. (21472 was also compared with the other Casella thermometers at 59°.5. The greatest discrepancy between the results and the Kew corrections was 0°.04.)

Whether it and all mercurial thermometers need appreciable systematic corrections between 32° F. and 100° F. to reduce their readings to those of an air-thermometer is not yet known. It is hoped to obtain this information at a later day.

§ 8. COMPARISON OF YARDS NUMBERS 6, 7, 8, 9, 10, WITH EACH OTHER.

For comparisons of the lake-survey yards Nos. 6 to 10 with each other and with Clarke yards A and B, the cylindrical ends of the brass yards rested in wyes, with a pressure of three or four ounces, nearly all their weight being carried by supporting spiral springs attached at one-fourth and three-fourths of the length of the yard. The comparators were firmly attached to strong timbers. The two yards under comparison were inclosed in a box with glass windows in its cover through which to read thermometers, and with slits with sliding covers through which the suspending wires passed, by whose aid, without opening the box, the yards under comparison were alternately placed in the wyes, which brought the centers of their ends in line with the axes of the sliding pieces of the comparators, and gave central contact for the end of the yard and the end of the sliding piece. The Clarke yards A and B remained in their cases, and were suspended and handled in the same way, the cases resting against adjustable stops, when exactly in the right position with reference to the comparators. Their cases were slightly inclined sidewise, so that these yards should slip so as to always take the same position in the case. Nearly all the comparisons were made in the cellar of the lake-survey office, where the daily range of the air temperature, caused mainly by the presence of the observers, rarely exceeded 20 or 3° F. Readings were usually taken about 94 a. m. and 4 p. m., the room at other times being kept carefully closed when the temperature of the cellar was below 40° F. The rise of temperature in the box containing the yards under comparison, from the presence of the observers, during the time (about ten minutes) required for two or three comparisons, was usually about 0°. 3 F., as indicated by the thermometers under the wrapping of three thicknesses of flannel which incased the brass yards. For higher temperatures the effect of the presence of the observers was less.

If the temperatures of the brass yards had risen by the same amount as the temperatures of their thermometers, namely, 09.3 F., the lengths of the yards would have increased by 0in.00010.

It it safe to say that they did not increase in length by more than one-half this amount, or 0.00005, which must have been nearly the same for both; so that there is no probability that, during the ten or fifteen minutes of comparison of two brass yards, the difference of their lengths changed by so much as 0.00002 on an average. As the yards were systematically alternated in position with reference to the ob.erver this error was eliminated in the result.

From the insignificant effect that this small rise in temperature would have on the difference in length of two similar brass bars, all comparisons made on two brass yards are used, until the thermometer has risen by 09.3 F. In early comparisons, at low temperatures, so many as five were obtained, occupying thirty minutes. Of these, only the first two or three have been used, as the thermometers rose 0°.6 or 0°.7 during the thirty minutes, and the temperatures of the two similar yards may not have changed equally.

In later work but two comparisons were usually made, each occupying from six to ten minutes, at 9.30 a. m. and 3 p. m.

In reducing the comparisons, those at the same period, and, therefore, at about the same temperature, have been combined so as to give a single equation of condition, to which a weight is assigned equal to the number of days of comparisons.

For this purpose, the mean of the observed differences of length is taken, and the

mean of the first readings of the thermometers in each set of comparisons, as the thermometers after the first reading are slightly affected by the presence of the observers, while the temperature of the yard has probably not sensibly changed. The form of an equation of condition for yards Nos.

and 7 will be

(6—7)t° — (6—7)26° — (62°—1°) (€6—er)

where 6 and 7 are the lengths of the two yards, to the observed temperature, and e̟ and e, the expansions of 6 and 7 for 1° F.

Or, placing (6—7)t° = n = observed difference of lengths; (6-7)52°=x=difference of lengths at 62°;

=

e-ey difference of expansions for 1° F.; 62°-to = b;

the equation of condition takes, for 6-7, the form

n−x + by = 0

For 68, 69, &c., x and y would receive one, two, &c., primes.

The first column of the following table gives the dates of comparisons, the second the number of sets of comparisons, the third the number of days of comparisons, or the weights, the fourth the total range in the observed values of n, the fifth, sixth, seventh, and eighth, the equations of condition, and the ninth their residuals. The unit is onemillionth of an inch. The values of b show how far below 62° F. the mean temperature of each group of comparisons was.

Solving the equations of condition by the method of least squares, and deducing the values of the unknowns and their probable errors, the following values result:

In determining these values, no comparisons of No. 8 with another yard made prior to August 7, 1875, are used, as that yard had previously changed its length.

Oin.00010

- Qin.00013 +0in.00020

Five days' comparisons, March 1, 1874, at 390 F., gave 6 - 8 Two days' comparisons, June 1, 1874, at 61°.2 F., gave 6-8 Two days' comparisons, July 20, 1875, at 64° F., gave 6 - 8 = In the interval between the last two comparisons, which show an evident change of length, as their difference far exceeds any possible error of comparison, the five yards placed end to end had been compared with the 15-foot brass bar.

During these comparisons the yards had been suspended by spiral springs, which allowed the end cylinders of the yards to rest in their guiding-wyes with but a few

ounces weight, and so left them with great ease of motion in the direction of their lengths. The idea at once suggests itself that in bringing the ends of the yards in contact, the shock of contact due the considerable mass of the yards had been sufficient to force in one of the agates at the ends of the yard No. 8 where the fitting had not been firm before, thus shortening the yard. These agates, in constructing the yards, were simply pressed into a cavity cut for them, and the brass at the ends of the cylinders burnished down around the agates. On August 2, 1875, I heated the ends of yard No. 8 to about 150 F., and then holding the yard vertical pressed the lower agate into a piece of soft wood, with a pressure of 20 or 30 pounds. Then the agate and end of the cylinder were rotated under the same pressure in a hole a little larger than the agate, the upper end of the hole having been reamed out so as to bear on the brass immediately around the agate and press it against the agate. Finally, the brass was burnished down upon the agate with a hand-burnisher. Both agates of No. 8 were treated in the same way.

Two days' comparisons, on August 7 and 9, 1875, gave, at 62°, 6—8=+0in.00012, which would indicate a lengthening of No. 8 by gin.00008 since July 20, 1875. But in the previous comparisons the comparing-room was visited three or four times during the day, which makes it uncertain whether the temperatures of the two yards were precisely the same.

As the burnishing could hardly have lengthened the yard, it seems possible that the small apparent change in length subsequent to July 20, 1875, was due simply to temperature errors in the previous comparisons, and that the length of yard No. 8 was not sensibly affected by the burnishing process, or, in other words, that its agates are now stable.

The comparisons of the other yards at different dates give no indications of change in lengths.

The following are the results of comparisons of Nos. 6 and 8 at different dates:

Oin.00000

Qin.00010

(in.00014

+0.00019

8

=+0in.00012

- 8

F., 1 day's comparisons, 6
F., 5 days' comparisons,
F., 2 days' comparisons,
F., 2 days' comparisons, 6
of No. 8 burnished down.
F., 2 days' comparisons, 6
F., 2 days' comparisons, 6
F., 14 days' comparisons, 6 8

+0in.00009
+0in.00011

1. Nov. 30, 1871. Temp., 55°.5 2. March, 1874. Temp., 39° 3. June, 1874. Temp., 62° 4. July 20, 1875. Temp., 64° August 2, 1875. Both agates 5. August 9, 1875. Temp., 62° 6. August 9, 1875. Temp., 63° 7. Dec., 1876. Temp., 41° The comparisons of November 30, 1871, were not made under temperature conditions which could make precision certain, and their discrepancy with the second comparisons does not make it sure that yard 8 had changed length in the interval. It certainly changed length largely between June, 1874, and July, 1875, and may have changed very slightly while being burnished, although the difference of lengths on July 20, 1875, and August 9, 1875, while larger than the probable, is not larger than the possible errors in the comparisons made on only two days.

§ 9. COMPARISON OF BRASS YARD NUMBER 6 WITH CLARKE YARDS a and b.

Having given the lengths and expansions of the other brass yards relatively to No. 6, it remains to explain how the absolute length and expansion of No. 6 were obtained from the Clarke yards A and B.

No. 6 was compared with A and B, and No. 7 with A. The yard A or B remained in its iron case, and, with its thermometers in their places, was put with the brass yard in the wooden box in which the brass yards had been compared with each other. The larger part of its weight was carried by suspending-springs, and its outer iron case rested against stops, adjustable so that the centers of its end surfaces should be in line with the axes of the sliding pieces or quills of the comparators. The case was slightly inclined, so that the yard within, under its own weight, should always assume the same position with reference to the case, and so with reference to the comparators. The brass yard was also suspended, resting its end cylinders in their supporting wyes. The two yards were alternately placed between the comparators, and comparator readings taken on both ends.

In the comparisons of No. 6 with A and B, at temperatures near or below 40° F., the three comparisons making a set required about twenty minutes. In this time the thermometer beneath the flannel wrapping of the brass yard usually rose about 0o,3 F., while those inside the cases of the yards A and B only rose about 0.005 F.

In comparisons near 62° F. the rise in thermometers was about one-half the above amounts. Hence, as the thermometer-temperatures of No. 6 and A or B change very unequally, and as their expansions are very different, only the first comparisons and the first temperatures, on entering the comparing-room after an absence of several hours, were used in the reductions.

No comparisons were used in which the thermometer in contact with the brass yard and under three thicknesses of flannel differed by more than 0.2 F. from the mean of the four thermometers inside the iron case of the Clarke yard; and as the presence of the observers for be two to four minutes before the thermomete s were read may have slightly affected the the mometer with the brass yard, while its effect on those with the steel yards was insensible, the thermometers with the latter yards have been alone used in fixing the temperature of comparisons. No comparisons were used where the thermometers, with the Clarke yards, changed by more than 0°.4 F. between morning and afternoon comparisons, (9.30 a. m. to 4 p. m.)

No. 6 and A were compared with each other at three different periods, at temperatures varying from 379.6 to 629.8.

No. 7 and A were compared at about 33°. As the length and expansion of No. 7, with reference to No. 6, are known with great precision, the comparisons of 7 with A were reduced to those of 6 with A, giving a fourth group.

Each of these four temperature-groups of comparisons gave an equation of condition of the form

(6—A) = (6 — A)62o — ( C2 — 1°) (€6 — ea ),

-

in which is the observed temperature, ea and es the expansions of No. 6, and A for 1° F., and (6—A)62°, and (e—ex), the unknowns, provided the temperatures were correctly measured.

But in some thermometers, which agree with an air-thermometer at 320 and 212° F., and which have no sensible errors of construction, it is known, in consequence of the dilatations of glass and mercury depending on both the first and second powers of the temperature, that at other readings they may differ sensibly from an air-thermometer. An attempt has been made to have one of the lake-survey thermometers compared with an air-thermometer, but as yet no results have been received. In the comparisons with each other of the brass yards Nos. 6, 7, 8, 9, 10, as their relative expansions, are very small, no sensible error is introduced into their relative lengths by an error of 09.1 or 0.2 F. in me uring their common temperature. But in determining the relative lengths of A and No. 6, an error of 0° 2 F., in determining their common temperature, would introduce an error of 0in.00002, a quantity larger than the probable error of comparison.

If such a systematic correction should be needed to make the lake-survey thermometers agree with an air-thermometer, it may be taken with sufficient accuracy between 320 and 620 F, as proportional to the excess of temperature over 32°, and its form will be A (to — 32°).

The equation of condition from a group of comparisons will then take the form

(6—A) to (6—A) 62° — { 52° — [ 1° + (1° — 32°) ▲ ]} - ex);

which may be written

―n+x-(ab▲) y =o,

in which n is the mean difference of lengths of the two yards, derived from a group of comparisons at nearly the same temperatures of which to is the mean; x is the unknown difference of length at 62, and y the unknown relative expansion. The weight of each equation is equ 1 to the number of its comparisons.

The following table gives the date of comparisons, the number of days of compari sons, the number of comparisons, the temperature-range during the comparisons, the equations of condition, and their residuals, ▲ being neglected. The unit is the millionth of an inch.

Date.

The following table gives similar data for comparisons of No. 6 and Clarke yard B:

In this table x and y' have replaced the x and y of the preceding table, as yard B has replaced yard A.

S lving the above equations by least squares, the following values result:

The unit being a millionth of an inch, and the probable errors being derived from the equations of condition when a is taken as zero.

If the value of x above be subtracted from that of x', we have A-B=152, the terms in a being neglected, as a is very small and its co-efficients nearly equal.

Referring to Colonel Clarke's values of A and B. § 2, we find A-B=211, differing from the value just found by 59. Both Colonel Clarke's and the lake-survey determination of this difference are indirect-that is, they are each obtained by comparisons with a third yard.

Still the difference is larger than would be supposed from the probable errors of the comparisons, and it is hoped hereafter to make a direct determination of its value. If equal weights were given to the two determinations of the difference, the two results would differ from their mean by but 120b000 of a yard, a quantity that is not large in such work.

Were the mean of these two values to be adopted, it would require a change in the values for the expansions of A and B, found by Colonel Clarke, of 3 part. Adding now the value of (6-A)6° to the length of A at 62° F, given by Colonel Clarke, (§ 2,) there results the length of No. 6 at 62° F.

No. 6=35.9995968-0.004408 A.

Adding the value of (6—B)62°, above, to the length of B at 620 F, given by Colonel Clarke, (§ 2,) there results the length of No. 6 at 62° F. :

No. 635in 9995387-0.004385 ▲.

The probable errors having, in both cases, been obtained by neglecting A as insignificant. These values differ by Qin.000058; a quantity larger than the probable errors of the separate values. The mean of the two values must be taken, and the probable error of the mean must be derived from its differences from the two values. There results, then, for the length of No. 6 at 620, from the data at present (April, 1877) available,

No. 6 35in 9995670in.000020-0.004396 A.

The probable error being obtained by supposing A =0, and being 100000 part of the length.

Residuals.