The error of the chronometer is found by comparing the time recorded by it when a star crosses the meridian, with the true time of passage of the star. The latter is tabulated in the Nautical Almanac under the heading of "right ascension." The observation of the moment of the stars crossing the meridian is made with a transit instrument. This instrument consists of a telescope mounted at right angles to a horizontal axis, which terminates in two cylindrical pivots, resting upon V shaped supports on a strong frame. In the field of view of the telescope is fixed a system of fine parallel threads which are seen distinctly with the image of the star, and over which the instant of transit of the star's image can be accurately noted. The telescope then being placed so as to revolve in the plane of the meridian, it is only necessary to note the time by the chronometer at which the star passes the centre thread. The difference of this time from the star's right ascension is the correction of the chronometer. This supposes the instrument to be in perfect adjustment, a condition never fulfilled in practice. The errors of adjustment of the transit instrument are: (1) The deviation in azimuth, the angle by which the horizontal axis deviates from an exactly east and west position. (2.) The error of inclination, by which the axis deviates from a perfectly hori zontal position. (3.) The error of collimation, or deviation of the straight line joining the optical centre of the object-glass with the centre of the system of parallel threads from perpendicularity to the axis. The effect of each of these errors upon the time of transit differs for different stars. Their angular amounts being first found, these effects are computed, and applied to the time of passage over the instrumental meridian, thus giving the time of transit over the true meridian. The azimuthal deviation is found by star observations, the error of inclination directly by the use of the spirit level, and the error of collimation by observing a terrestrial object or a slow moving star in the direct and reversed positions of the instrument, reversal being effected by turning the axis end for end in the V's. Or, the latter error may be found along with the error of azimuth by solution of the observation equations of a number of stars taken in both positions of the instrument, as will be shown below. The angular values of the three errors being denoted by a, b and c, respectively expressed in seconds of time, their effects on the times of transit are Aa, Bb, and Cc, where Asin (-8) sec 8, B= cos (4-8) sec S, C=sec 8, being the latitude of the place and & the declination of the star. If then a be the right ascension of the star, 7 the chronometer time, and its correction, the observation equation is AT+Aa+Bb+Cc-a-T b, and therefore Bb being known by the sliding level, this equation contains three unknowns, T, a, and e, which can be found from three such equations by the ordinary algebraic solation, or from a greater number by combination according to the method of least squares. This is the method used with the portable transit instruments supplied to our observers in 1885 and 1886, one of which was again used by Mr. Klotz in his obser vations this year. The other one was taken by Mr. Ogilvie on the Yukon River expedition. Another instrument of this kind not being procurable on short notice, I was obliged to fall back upon an instrument of a different pattern, an altazimuth which Mr. Russell had constructed some years ago for triangulation work. This instrument I used as a transit instrument, but its theory presents some points of difference from that of the ordinary transit instrument. It is an altitude and azimuth instrument having a horizontal limb 17 inches in diameter, read by two microscopes to 1", and an 18-inch vertical circle, read by 4 microscopes each to 1". The telescope of 31 inches aperture, and 36 inches focal length is, in order to avoid unnecessary height and consequent instability, attached. to one end of the axis beyond the standard. The value of one division of the strid ing level is 1"-38 or 0s 092. The upper or revolving horizontal plate is supported on the lower by three small rollers attached to the upper plate and running on the rim of the lower plate. The two plates are clamped together by means of a collar clamp on the short central axis. For very firm clamping during transit observations, there is an arrangement for clamping the rollers. This, however, I found unnecessary, as the instrument when the collar clamp only is applied has, owing to its weight, remarkable stability in azimuth. The whole instrument is supported on a well-braced tripod with a brass head. The instrument when clamped in the meridian is used for transits in the usual way, but from the position of the vertical circle microscopes, it is impossible to reverse the axis end for end, which process, in the ordinary transit instrument, gives the collimation error with opposite sign without changing the azimuth. Reversal of the collimation must therefore be performed by revolving the horizontal plate of the instrument 180°, resetting by means of the reading microscopes. This however, cannot be done with perfect accuracy and necessarily changes the azimuth. There exists moreover in this instrument a large inequality of pivots, whereby the striding level reading of the inclination of their surfaces does not give the true inclination of the axis of revolution which is the line joining the centres of the pivots. This error which is found in the transit instrument directly by striding level readings in the two positions of the axis in this instrument can only be found indirectly. There is also a flexure of the axis, caused by the weight of the eccentric telescope. This has the same effect as an inequality of pivots. The combined flexure and inequality of pivots, if not directly determined by proper methods, enters into the observation equations as an unknown level error, so that in working out the observations, the term Bb must be retained. One right's observations usually consisted of twenty stars, five in the position telescope east, then five telescope west, then five again telescope west, and five telescope east. The observations of each star are reduced to the mean thread, and -corrected for the rate of the chronometer to one instant, and for the inclination of the axis as given by the striding level, and for approximate values of the azimuthal deviation, inequality of pivots, collimation error and chronometer correction. We have then for determining the resident errors of azimuth, level, collimation and chronometer correction, for each star an observation equation: Aa+Bb+Cc+&t=d. If the star has not been observed over all the threads, a weight correction should be applied. Five threads making a perfect observation, I used weights, p=0.9, 0·8, 0-64, and 0.4, for four, three, two and one thread respectively. The equation then becomes: '3' pAa+pBb+pCc+p&t=pd. For each set of five stars there is a different a, so that there are seven unknowns ɑ ɑ ɑ ̧, a, b, c, d to be determined from the twenty equations. b and c have opposite signs telescope east and telescope west, so that B and C are given negative signs for telescope west. Multiplying each equation by the coefficient of a in it and adding we obtain the normal equation in a, similarly for the other unknowns. The normal equations then are : (1) [p A;] a, (2) (3) (4) [p Az]a, 3 3 2 +[p 4, B] b+[p A ̧ C] c+[p 4,] dr=[p A ̧d] [p A]a,+[p 4, B]b+[p 4‚C] c+[p4,] òt=[p 4d| (5) [p4,B]a,+[p4, B]u,+[pA ̧B]a ̧+[pa‚B]a,+[pB2]b+[pBC]c+[pB}òt=[pBd] (6) [p4,C]a,+[pA ̧C]a2+[p4 ̧C]a,+[pA‚C]a,+ [pBC]b+[pC2jc+[pC2]òr=[pCd] · (7) [p 4,] α, +[p ^2] a2+[p4,]a ̧+[pa‚]a ̧+[p B] b+ [ p C] c+ [p] dr=[pd] In the calculation of the coefficients of these equations, it is necessary to find for each star thirteen quantities viz., A, B, C, d, A2, B2, C2, BC, CA, AB, Ad, Bd, Cd, which are set down in thirteen columns, each quantity being multiplied by the weight of the observation. These columns summed give the coefficients of the normal equations. Any column containing A, as a factor, i. e., A, Ad, A2, AB, AC, must be summed separately for each set of five stars. The normal equations are then solved by the method of substitution, i. e., equation (1) is divided throughout by the coefficient of a,, [p A;],and multiplied by [p A1b] and then subtracted from equation (5) thereby eliminating a', from that equation. Then equation (2) is divided by [p A and multiplied by [p A, B] and subtracted from the remainder of equation (5). Similarly the equations (3) and (4) are divided each by the coefficient of a, and a, in it and multiplied by the coefficients of these quantities in equation (5) and subtracted from (5). We thus get in place of equation (5) an equation clear of the unknowns a, a, a,, and a, and containing only b, c, and or as unknowns. Equations (6) and (7) are treated similarly, and we then have three equations each containing b, c, and S. Successive operations of the same kind eliminate b and c and give the value of St. Substituting this value in the last equation but one, which involves c, and St, we have the value of c, thence by successive substitutions we obtain all the unknowns b, a1, a2, az, α. Each of these should be found from the equation in which its coefficient is greatest, and then we have a final check on the whole by substituting the values found for all the unknowns in the normal equation in ST (No. 7), which ought to reduce to an identity. For the sake of readiness in the computation the successive coefficients of the unknowns are placed in columns, and a check on the work is carried through by placing in an additional column, the sum of the coefficients of each equation with its sign changed, which sum is subjected to the same processes of multiplication, division and subtraction as the other coefficients of the equation, so that at each stage of the process the sum of the quantities in a line must be zero, if no arithmetical mistake has been made. For the sake of illustration, I subjoin the working out after this method of observations taken at Winnipeg on 4th October, 1887. There are seventeen stars this night in four sets. One or two of the transits were defective, rendering necessary the application of weights. The column headed is the chronometer time of transit over mean thread, corrected for rate, and for approximate azimuth, inclination, and flexure errors a is the star's right ascension, then a-τ gives approximate chronometer correction. Taking -4h 02m 598.204 as a near approximation to the actual chronometer correction at the time 2h 00m 008, to which time the observations have been corrected for the rate, the differences between 4h 02m 59s 204 and the quantities a- are tabulated in the column d. The star numbers in the first column refer to the Berliner Jahrbuch from which the stars' right ascensions are taken. |