by the tabular difference corresponding to one second for the angle A', in a table of logarithmic sines, calling the tabular differences = log sin B' + dvi yvi + log sin B' + ¿vii vii which may be written in the form 2. We have now shown how to obtain the numercial equations of condition from rigid geometrical ones. As there is sometimes difficulty in selecting the latter, some remarks may be useful, as, if too few are selected, the adjustment will be incomplete, and if too many, (some not being independent,) the labor will be unnecessarily in creased. In a continuous triangulation, one measured base and two measured angles in each triangle will absolutely fix all other parts of the triangulation by computation. If any of those other parts are also measured, each such part will give an equation of condition. Side-equations. From a known side two directions fix a third point. From any of these three points two directions fix a fourth point, and so on. In a net of p points, of which two are known, being the ends of the base, p-2 will have to be fixed and will require 2 (p-2) directions from known points. Any additional direction observed from a known point will be superfluous and will give a side-equation. If the p points be connected by l lines observed over from a known toward the unknown, we shall have, excluding the base, 1-1 directions, of which 7-1-2 (p-2) are superfluous, giving 1-2p+3 side-equations. Angle-equations.—In a polygon of p sides, the sum of the angles is known, and hence gives an equation of condition. If two of its vertices be connected by a new line, two new polygons will be formed, giving two new known sums, but only one independent equation of condition, since one of the sums depends on the other and the original sam. Hence each new line which must be observed over from both ends gives an angleequation. When there are lines there will be 1-p+1 angle-equations. The number of side and angle equations now being known, to find them proceed as follows: Starting from known points, two directions toward a new point fix it; and each additional direction from a known point gives a side-equation; if, at the new point, the independent angles between these directions are observed, each such angle gives a sideequation. APPENDIX D. C. B. C. LENGTHS OF THE FIVE NEW LAKE-SURVEY YARDS. The five new lake-survey yards are marked L. S. No. 6, 7, 8, 9, 10. They are brass bars, one inch wide by sixth-tenths of an inch thick, having cylinders four-tenths of an inch in diameter at each end, bearing agates ground to a radius of four inches. The extreme points of the agates are approximately one yard apart, and project with their cylinders six-tenths of an inch beyond the ends of body of yards. The two new yards of the Office of Weights and Measures, with which the lake-survey yards were compared, are called "Transfer Yards A and B," and are like the former in every respect, save that at the middle of their lengths a small well is bored to the middle of their thickness, and at the bottom of the well there is a fine line engraved on a small silver surface. The object of this line is the ready microscopic comparison of the lengths of these two end-measure yards with the British standard-yards, bronze No. 11 and malleable iron No. 57, in possession of the Office of Weights and Measures, these latter being line-measure yards. The microscopic comparison of No. 11 with the distance between the two lines in the wells of A and B when they are arranged in line and in contact, in the four different possible orders, gives at once the lengths of A and B in terms of No. 11. In determining the values of A and B in terms of British bronze No. 11, (A+B,) was determined on the new microscopic comparator of the Office of Weights and Measures, by Mr. Lane, and Mr. Hilgard gives as the probable error of the resulting value of (A+B) 0.000014 inch. The value of (A-B) is 0.000891 inch, and was determined on the Saxton's pyrometer. Mr. Hilgard does not attempt to assign the probable error of this value, which must be much less than that of (A+B). Mr. Hilgard gives as the result of comparisons of transfer-yards A and B with British bronze No. 11, the temperature being 61.79 F, at which No. 11 is a British yard. A No. 11+0.0009197 inch. Taking the co-efficient of expansion of A and B as 0.0000097, which Mr. Hilgard thinks it does not exceed, we have for lengths of A and B at 62° F., denoting one British yard by Y. A at 62° FY+0.000993 inch. B at 62° FY+0.000102 inch. The comparisons of L. S. yards Nos. 6, 7, 8, 9, 10, with A and B were made on the Saxton's pyrometer of the Office of Weights and Measures, at temperatures varying from 55 to 640.5 F. They are made of the same material as A and B, and their comparison with our 15-foot bar shows that they differ in co-efficient of expansion very slightly, if at all, from the value given above for A and B. JOY If we assume their co-efficients of expansion the same as those of A and B within the range of comparison, the following values for their lengths result, from the comparison I thus far made. These results must only be considered as preliminary, as further comparisons will be made. Lake-survey yards Nos. 6 and 7 were compared with each of A and B, and on more Ps than one day and combining the comparisons by least squares, the probable error of the resulting values of 6 or 7 in terms of A or B is found to be 0.00001 inch. L. S. yards 8, 9, and 10 were less carefully compared, and will be again compared with Nos. 6 and 7. Mr. Hilgard's letter is appended: "UNITED STATES COAST SURVEY OFFICE, Washington, April 22, 1872. "DEAR SIR: I am at last enabled to give you the result of the comparison of the two 'well' yards, or transfer yards' as we have now called them, with the British standard. "The one formerly designated as 101 we have called A. "We find that A+B twice the British bronze yard No. 11+0.000948 in. at the temperature of 610.8 at which the latter is standard. From a comparison of the 74 complete observations made, at temperatures ranging from 42° F. to 73° F., including 20 near the standard temperature, a probable error of ±0.000014 in. may be ascribed to this value, or ±0.000010 in. for one yard. I would not, however, draw the limits of precision so close until verified by observations in different years. "The comparisons of A and B were made on the pyrometer, the bars being supported in precisely the same manner as in the line comparisons. Observations were made on thirteen days at temperatures ranging from 500.1 F. to 62°.9 F. No difference depending on temperature could be discovered. "The contacts were re-adjusted every day, and the ends interchanged. The probable error of one day's result is but Togo of an inch. I do not attempt to assign that of the mean value, which is A-B=0.0008912 in. "We have, therefore, A=Br. yard No. 11+ 0.0009197 in. B=Br. yard No. 11+ 0.0000285 in. "Full experiments for the dilatation of the brass yards remain to be made. The following data may be useful in the mean time. The comparisons with the bronze yard above referred to show an increase in the length of the brass yard over that of the bronze standard of 0.0000049 in. for 1° Fahr., and as the expansion of the bronze bar is stated at 0.000342 in. for each degree, that of the American yard becomes 0.0003469 in. "Similarly we have found from comparison of between temperatures 610 and 80° the iron meter with a brass one of the same metal as the yards, an increase of expansion of 0.0001301 in. over that of the iron for 10; taking the proportional expansion of the latter at 0.00000642, we find for that of the brass bar 0.00000973. The preceding value corresponds to 0.00000964. Mr. Hassler found 0.00001051, but in order to bring Mr. Hassler's value of the meter observed at 32° to Clark's value at 62° we must allow a proportional expansion of 0.00000962 for the Troughton scale, the expansion of which has not been found to differ visibly from that of our yards. It appears probable that the co-efficient of expansion of the yards does not exceed 0.0000097. "Yours, respectfully, "J. E. HILGARD, "Assistant in Charge. APPENDIX E. OFFICE UNITED STATES LAKE SURVEY, January 16, 1872 MAJOR: In obedience to your orders, I have the honor to submit the following report upon the constants and errors of the new Troughton & Simms repeating theodolite. This instrument has a horizontal and a vertical limb, each 12 inches in diameter, and graduated to five minutes. The horizontal limb is read to single seconds by two opposite reading-microscopes. The vertical limb is read to five seconds by two opposite verniers. The telescope has a 2-inch object-glass, and a focal length of 18.7 inches. It is provided with an illuminating axis-lamp, one diagonal and two direct eve-pieces The outer vertical axis carries the horizontal limb, is 4.2 inches long, and has a diame ter of 1.3 inches at the upper, and 1.2 inches at the lower end. It fits into a tripod, and rests upon a shoulder at its upper end. The shoulder is inch wide. There is no means of adjusting this axis so as to make it fit tighter or looser in the tripod. At present it fits perfectly. Both axis and tripod are of brass. The inner axis carries the telescope and reading microscopes, is of steel, 5 inches long, diameter at upper end Å inch, at lower end inch. The lower end rests upon an adjustable plate, so that it can be raised or lowered. This axis was broken last summer while the instrument was being transported from this office to Green Bay, Wisconsin. A new one has been made by Würdemann. During the examination the instrument was mounted on one of the stone posts in the observatory. The value of a division of the level on the vertical limb was found by comparing it with the level belonging to astronomical transit No. 15 (Würdemann.) The mean of 44 level readings gave a value of 2".24. The same result was obtained by compar ing it with the vertical limb. The value of a division of the axis or striding level was found in the same way. The mean of 36 level readings gave a value of 1.05 for one division. The angle which the two axes make with each other was found in the following manner: The inner axis was clamped to the outer one and the level read. Both axes were then revolved 180° and the reading repeated. This eliminates the error of level and gives the inclination of the outer axis in the direction of the level. This was repeated ten times and a mean taken. The outer axis was then clamped to the tripod and the inclination of the inner axis found in the same way and for the same direction. The difference of these two inclinations is the angle which the axes make with each other in the direction of the level. The level was next placed in a new position, 120° from the first, and the operation repeated. If the outer axis is perpendicular, then these differences are the deviations of the inner axis from the vertical. If the level be at right angles to the inner axis, these differences are the deviations of the level from a horizontal plane. The place where the deviation is a maximum and the maximum deviation can be computed by the following method: C D Q B |