the base by reference to the mercurial temperatures read during the comparisons and measurements.

This method has indicated, with what seems to be considerable weight, a change in length of one of the wires during the measurement of the base line, and permits of pretty accurately locating this change. But it is not considered that a mercurial thermometer can be closely relied upon to give the temperature of a wire beside it, especially in the sunshine, as when some of the measurements of the base line were made, and while the probable errors indicated are quite small, it is not likely that the absolute length of the base is as accurately known as these probable errors would indicate. This fact, however, remains true for other systems of measurement, as constant or systematic errors may exist which can only be revealed by remeasurement with different apparatus and methods.

In obtaining the equations of the wires the observations are treated in groups of five, the mean being used as one observation with weight unity. Each group then gives an observation equation of the form

==

x + (t − to) y = l

where x = length of wire minus the length of comparator I at a temperature to, y= change in length of wire for 1° C. change in temperature,

t

temperature for any group of observations,

to an initial temperature (actually taken at about the mean temperature of the comparison for the given section), and

observed value of wire length minus comparator interval.

Each section of each wire has from 30 to 40 observation equations of this form, some of the temperatures being high and some low. From these are obtained the normal equations

and [t―t] + [ (t − t) 2] y — [ (t − t) 1] = 0 for y,

a solution of which gives x, the difference in length between wire and comparator at temperature t, and y the change in length of the wire for 1 degree centigrade. The value of x and y being abtained for the ten sections of each wire, it remains to combine them to obtain the total length of the wire and the mean coefficient of expansion. To obtain the total length of a given wire, it is only necessary to reduce all of the values for x to a common temperature tw, by applying the coefficient of expansion for the difference between tw and to of each section and then add the resulting values of x. To obtain the most probable value of the coefficient of expansion of a wire, either of two assumptions may be made. It may be considered that the wire is actually nonhomogeneous and that the several sections have different coefficients of expansion, or it may be considered that the variations in the coefficients among the sections is due to errors of observation.

By plotting the values of the coefficients for the several sections of the three wires it is found that they give very similar curves-that is, when a section of one wire has been found to have a low coefficient the corresponding sections of the other wires have also a low coefficient. The curves for steel wires Nos. 1 and 3 follow each other very closely, indeed, while wire No. 2 departs from the others in two sections where the weights of the determination are low. This points very strongly to the conclusion that the assumption of nonhomogeneity is untenable, since there is no apparent reason why the coefficients of corresponding sections of different wires should agree more nearly than the several sections of the same wire. It is very much more reasonable to suppose that the thermometer did not give the actual temperature of the wires, and this would account for the agreement of coefficients obtained for the sections compared with the comparator interval at the same time, since the same temperatures were, of course, used in reducing the results for all the wires. It is therefore considered that the weighted mean of the results for the several sections will give the best value for the coefficient of expansion of the whole wire, and this method was followed in obtaining these coefficients.

In the determination of x and y for each section of the wires there are n observation equations and two unknowns. The probable error of a single observation of weight unity is 0.6745), and the probable error of the resulting value of a [vv] is the above divided by the square root of the weight of x, or 0.6745

where [p] equals the number of observation equations, since p is unity; the probable error of the resulting value of y is 0.6745

[vv]
[(t-t)2] (n-2)'

The probable error of x for the total wire length is simply the square root of

the sum of the squares of the probable errors for the several sections. The probable error of the coefficient of expansion y for the whole wire is 0.6745

[pvv] [p] (n-1)'

p in this case representing the weights of the several values of y obtained from the normal equations.

The following equations are thus obtained for wires 1 and 2, where L, and L2 -the lengths of wires 1 and 2, respectively, I=comparator interval, and t-temperature of the wire:

and

L, 10 I-284.780.12+(t−2.5° C) (11.147±0.054)

L, 10 I-382.33+0.11+(t-2.5° C) (17.126+0.034). Introducing the value of I as already given, I=100.009720 meters+39 microns, we obtain the wire equations

L,

mm.

999.81242 meters.00041 meters+(t-2.5° C) (11.1470.054) L=999.71487 meters.00041 meters+(t−2.5° C) (17.126±0.034)

THE LENGTH OF THE MACKINAW BASE LINE.

Having arrived at the length of kilometer wires Nos. 1 and 2, we can now give the length of the Mackinaw base line in terms of these with which it was measured. Previous, however, to giving the results a brief description of the method of preparing the base for measurement will be given. The line having been cleared of all obstructions and a station erected over west base, from the top of which a target centered over east base could be seen, the line was ranged through with a transit, and a hub, with a tack to mark the line, set every 100 meters. For measuring these 100-meter distances a No. 18 steel wire, marked by brass beads into 10-meter intervals, was used. After the hubs were all set the 100-meter wire was stretched between consecutive hubs and the elevation at each 10-meter point obtained by a line of levels, the rodman holding his rod at each bead. These elevations were then platted on profile paper and a profile of the line obtained. On this profile the grade line for each kilometer section of the base was established, and from it the length of each stake for supporting the wires was taken.

A kilometer section was made ready for measurement by again stretching the 100-meter wire between consecutive hubs and setting on the line with the aid of a transit, at each 10-meter mark, a small stake about 2 cm. square by 20 cm. long, driven approximately 15 cm. into the ground. These stakes, with the hubs, located the position of each tall stake for supporting the wires. The tall stakes having been cut, sharpened, and distributed along the kilometer, were set as follows: One man passed along with an iron bar and made a hole in the ground opposite each stake or hub, and about 20 cm. from it, measured at right angles to the base. This man was followed by three men, furnished with a stepladder and wooden maul, who drove the tall stakes, taking care to see that they were all approximately plumb.

On each of these tall stakes the grade line was established by setting with a level or transit, as was most convenient, a sixpenny nail, driven at the proposed elevation of the measuring wire. The supporting hooks, previously described, were next fastened to the tall stakes by a couple of men passing along the line with a hammer and stepladder, hooking a set of hooks under the grade nail, pulling the strings taut, and driving the large nail into the stake on its face side at right angles to the line. When the hooks were not in use supporting the wires they were kept hooked under the grade nail to prevent their being whipped about by wind. The two men setting the hooks were followed by two others, who hitched a plumb bob to the middle supporting hook and plumbed its point of support over the line stake, by tamping the earth in the proper manner around the tall stake. The measurement of a kilometer took place as follows: The reels containing the measuring wires were placed tandem about 5 meters back of the mark from where the measurement was to begin and securely fastened in place. The ends of the three wires were run out and tied to a short stick in the order in which they were used in measuring. A man took this short stick in hand and marched forward with the wires, the reels in the meantime being tended to pay them out evenly. When 200 meters were out the man pulling the wires stopped, and a couple of pieces of wood about 12 cm. long by 3 cm. square, held together at one end by a screw, were clamped onto them, notches having been cut in one of the sticks to receive the wires. A second man took hold of this clamp, when a signal was

given to move forward, and 200 meters more were run out, when a second clamp was put on and a third man took hold, and so on until the wires were out. The rear ends of the wires were then fastened in their respective clamps and a signal given to put them up in the supporting hooks. At this signal each man started from his clamp and began hooking up the wires, working toward the rear end of the kilometer. As soon as the wires were all hooked up, each wire was fastened to its respective spring balance and the proper tension applied and maintained, one man devoting himself to this adjustment. The attendants, on the signal being given that the tension had been adjusted, began plumbing the supporting hooks, first passing over the line and plumbing them approximately by eye, then following with a small weight of about one-half kilogram, which was hung to a supporting hook, the wire lifted out and returned when the hook had been brought plumb. It was customary to plumb through the length of the kilometer six times before starting a measurement, although it was found by trial that only four times were actually required to bring about a uniform tension in the wires.

The plumbing of the supporting hooks being finished, a flag was hoisted on a small pole at the forward end of the kilometer as a signal that on the next even minute observations would begin. All watches had been previously compared, and on the even minute the three scales were read as rapidly as possible, the proper tension on each wire having been noted. The reading of each thermometer was also taken and recorded, the observer at the rear end of the wires in the meantime having seen that the rear marks were in place. After about 20 readings had been taken at one-minute intervals, or earlier than this if the wires had expanded or contracted as much as 1 centimeter, observations were discontinued, and the flag that had been raised was dropped as a signal that the supporting hooks were to be replumbed. Each attendant would then plumb backward and forward over the 200 meters that he cared for. This replumbing finished, the observations proceeded as before, a whole day being devoted to the measurement of a single kilometer.

The Mackinaw base line is approximately 6.6 kilometers long and was measured in seven sections. The first six sections were each approximately 1 kilometer, and were measured, using the full length of wires Nos. 1, 2. and 3. The seventh section was approximately six-tenths of a kilometer long, and was measured with the first six 100-meter sections of wires Nos. 1, 2, and 3. The end of this seventh section reached a point approximately 19 meters beyond east base, the distance from the end of the seventh section back to east base being measured with a fractional part of a 100foot Chesterman steel tape.

COMPUTATION OF THE LENGTH OF THE MACKINAW BASE LINE.

In reducing the notes for a given kilometer of the base line the observations were treated in groups of 5, the mean of the 5 scale readings being considered as one observation. As 5 thermometers were read, the mean of 25 thermometer readings gave the temperature to use for each group of 5 scale readings. The groups of scale readings and corresponding temperatures were tabulated and the mean temperature for the whole kilometer was obtained. The residual of each group temperature from the mean was then taken out and the scale reading corrected to the mean temperature by applying the product of this residual and the coefficient of expansion of the wire under consideration. The mean of these corrected scale readings subtracted from the length of the wire at the mean temperature gives the distance from the point where the rear beads are held to the zero of the scales. The corrected scale readings mentioned above serve to determine the probable error of the comparison of the wires with the kilometer.

The results of this computation for the several kilometers are given in Table No. 5:

TABLE 5.

Mean

Mean scale readings at
Date of Number tempera- mean temperature.

Number of kilometer. measure of obser- ture at

ment.

vations.

which reduced.

In measuring the base line the tension was applied to the wires by means of spring balances, while in determining the lengths of the wires on the comparator in 1897 weights were used which were attached to wires passing over the tension wheels, as already described. In the measurement of the base the tension exceeded that used on the comparator by 7.9 ounces for No. 1 wire and 9.1 ounces for No. 2 wire. The effect of increased tension was determined by direct experiment on the base line and found to be 0.709 mm. per ounce for No. 1 and 1.265 mm. per ounce for No. 2. Since the wires were longer on the base as the result of the increased tension, the resulting length of the base is too short and a positive correction is applied, as indicated in columns 3 and 4 of Table 6.

The elevations at the several kilometer points were referred to the water surface of Lake Huron and thence to sea level. This gave the data for grade corrections and reductions to sea level given in columns 5 and 6. The corrected lengths for the base are given in columns 7 and 8.

It is seen from Table No. 6 that the lengths of the several sections of the base as derived from the two wires do not agree. This might be due to several causes: First, to imperfect fieldwork; second, to an erroneous value for the length of one or both of the wires; or third, to erroneous coefficients of expansion. If these differences were due to the first cause mentioned, it would not be expected that

they would follow any fixed law. If the third assumption were true, the differences would vary with the temperature at which the sections were measured. In Table No. 7 these differences in the computed lengths of the sections are taken out and it is seen that they are very nearly constant for the first four sections at about 31 mm. and that for the fifth and sixth sections and the remeasurement of the first the difference is about 27 mm. It is also seen that the two measurements of the first section differ by 0.8 mm. for wire No. 1, and by 4.2 mm. for wire No. 2. The quantity 0.8 mm., so far as it goes, indicates a shortening of wire No. 1, but it is so small that it may be attributed to errors of observation and it is so treated. The sudden change in difference between the fourth and fifth kilometers indicates that wire No. 2 was stretched about 4 mm. between the measurements of these two sections. The value of this amount of stretch is indicated to be 4.2 mm. by the variation in the length of the first kilometer as obtained before and after the measurement of the other sections of the base. As the lengths of the wires were determined in 1897, after the measuremeut of the base, kilometers 1 to 4 should be corrected for this stretch. This has been done, giving the results in columns 7 and 8, and the differences are then nearly constant, as shown by coluinns 9 and 10.

As to the cause of this difference between the results by wires 1 and 2 no explanation can now be advanced. It is apparent that the absolute lengths of the wires used in the computations were not the lengths existing at the time of measuring the base. It is proposed to go over the computations carefully at the first opportunity to make certain that no error has been made in deducing the lengths of the wires from the comparator comparisons. It is also proposed to work up the wire lengths in terms of the difference in length without recourse to the mer curial thermometers. If the resulting wire lengths are found to be essentially correct, the only explanation left seems to be that the wires actually changed in length between the base measurement and the comparator work. Some weight is lent to this theory by the fact that the few observations obtained with the wires on the comparator in 1896 do not agree with the results of 1897. Since the 1896 comparisons were imperfect, some of them being made when the conditions were known to be poor, as it was necessary to get into the field as early as possible, they have not been considered in determining the length of the base. It is proposed, however, to work them up carefully and give them the weight to which they seem entitled.

We may now give the length of the base by the two wires, as obtained from the present computation, Table No. 8. The apparent probable errors given in the table are deduced later.