S17 



TRANSIT, OR TRANSIT INSTRUMENT. 



TRANSIT, OR TRANSIT INSTRUMENT. 



318 





with a micrometer to his transit, he proceeds exactly as we have 

 described above to measure the quantity and direction of the collima- 

 tion, the quantity and direction of the inclination, and then observes 

 away, only taking care to note as many standard stars as will give him 

 a correct clock-error, and at least some so far apart as will enable him 

 to detect his azimuthal error. If he observe one or two stars near the 

 pole, so much the better. Now calling the correction for the clock 

 at the mean of the time of the observations of standard stars ; t,t?s", 

 4c., as before the observed times of transit, corrected for collimation 

 and level as aforesaid ; p, p', p", &c., with their proper signs, the 

 correcting factors of the unknown azimuth x ; and a, a', a", &c., the 

 apparent right ascensions of the standard stars, he has the following 

 series of simple equations : 



+ +px = o 

 if + e + px' = a' 

 " + t+px" = a" 



&c. &c. &c. 



Group together the equations in which the coefficient of x is nearly of 

 the same magnitude and with the same sign,* and dividing each group 

 by the number of its component parts, so as to have e with unity as a 

 coefficient, form at least two equations in which the coefficients of x 

 differ considerably, and subtracting one from the other, a value of x 

 will be found with its proper sign. Substituting this value of x in 

 each of the equations t = a i px, you will have as many values of e 

 as you have equations; and taking the mean of those which are 

 derived from the quick-moving stars, you have a good clock correction. 

 Now if any other objects have been determined, the right ascension of 

 which is required, add (speaking algebraically) the sum of all the 

 corrections for collimation, inclination, declination, and clock-error to 

 the observed transit, and you will have the apparent right ascension. 



But it most frequently happens that the observer has no collimating 

 mark and no micrometer to his instrument ; nay, he may only have a 

 view out of a window t which commands no distant or distinct object, 

 and not be able to see even the zenith, much less the pole. This last 

 is the greatest objection ; for the accuracy of the meridian adjustment 

 depends chiefly upon getting stars near the pole. To detect the error 

 of collimation the observer must proceed thus: After having carefully 

 determined the inclination of the axis, he observes as many well-known 

 stars as he can, especially getting them as high and as low as possible 

 for ascertaining azimuthal error. He must then reverse the axis, 

 repeating the measurement of the inclination, but by no means 

 touching the elevating screw of the instrument, and make a similar 

 set of observations. A series of equations must be formed for each 

 position of the instrument, which will be of the following form : 

 supposing 6, t' *'', to be the observed transits of the stars corrected for 

 inclination imlij ; q, </ q" , to be the values of the secant of declination 

 for each star respectively ; r, the unknown value of the collimation 

 error in the first position (which becomes c when the instrument is 

 reversed; ; x, the error of deviation ; and t the clock correction, 



and so on, for the stars first observed, and 



i" + - q"c + p"x = o" 



and so on, for the stars of the second set after the instrument is 

 reversed, 



The mode of treating these equations will differ according to 

 circumstances. They might be solved by the method of least squares ; 

 but it is scarcely worth while in ordinary cases to use any such refine- 

 Form four groups, two in each set, those in which x hax the 

 largest, and those in which s has the smallest coefficients, dividing each 

 group by the number of it* component parts, so as to leave < with 

 unity for its coefficient. (Jail these equations 1, 2, 3, and 4. Sub- 

 tracting (4) from (1), we shall eliminate t, c will have a + coefficient 

 exceeding 2, and ./ in the most difficult case, that in, when the observer 

 ily look to the south, has a small positive coefficient. Again 

 subtracting (3) from (2), we shall have c with a positive coefficient 

 exceeding 2, and x with probably a small negative coefficient. From 

 these twq equations c can be determined pretty accurately. Substitute 

 this value in equations (1) (2) (3) and (4), group (1) and (3) together, 

 and (2) and (4) together, and we have a pair of equations in which x 

 has coefficients considerably unequal ; and by subtracting one from the 

 other, c is eliminated and ./ determined with tolerable accuracy. 

 1- MI illy, the substitution of these values of c and x in the original 

 equations will give a satisfactory clock-error if the observations are 

 gtxxl and prgtty numerous, even although the observer has not more 

 than 50 of clear sky to work upon. The times of transit of other 

 objects must be corrected by the quantities thus found, and in this 



* If two known circumpolar starn, like > I'rsto and Coptic! 51 Hevelil, are 

 observed, subtract one of these equations from the other, and you have an 

 equation in which z ha* a large coefficient, and therefore a good determination ; 

 If I'nlarlx U well observed, use ft singly, and group the standard stars together 

 which hare not more than SO" or 40' declination. 



t This was the cac with Homer, and the transit Instrument was Invented 

 prcciwly for neh > situation, lee Buls Astronomla,' cap. 8. 



way apparent right ascensions may be deduced with considerable 

 certainty. 



The clock correction should evidently come out the same in both 

 positions of the instrument, and the differences from the mean fall 

 within the ordinary errors of observation. If this is not the case, and 

 there should be re,ison to fear any alteration in the position of the 

 stand or the Y'S in the process of reversing, the values of x cannot be 

 assumed to be the same in both groups. If the time should be 

 required with extreme accuracy from such imperfect observations, the 

 observer may alter the quantity of collimation in his calculation till he 

 does get the same clock-error, although with different deviations, from 

 both sets. This may be done by one or two trials, but generally 

 speaking the mean of the clock-errors from both sets will be near 

 enough, and not differ sensibly from the more elaborate calculation. 

 It is not however easy to get the time very satisfactorily without being 

 able to see the pole, or at least the zenith. 



In what precedes we have supposed the extreme case, that is, that 

 nothing is to be seen north of the zenith, and that x therefore has 

 always the same sign. The intelligent reader will be guided in 

 practice, not by the directions here given, but by the value of the 

 coefficient* of his utdnoim quantities, a discretion which some astrono- 

 mers cannot or will not use. 



It is always desirable that the value of the three transit corrections 

 should be small (indeed the formula! are not exact, when the errors are 

 large), to save unnecessary trouble in multiplying. The method of 

 measuring the inclination implies that you can rely on the scale of the 

 level for the quantity measured, which is scarcely true when the 

 amount exceeds a few seconds of space. The collimation error is 

 easily brought within reasonable limits, if the observer has a micro- 

 meter, or can see any fixed object distinctly while he alters the screws. 

 The azimuthal adjustment requires either an object of reference, 

 which is always the case in principal observatories, or adjusting-screws 

 of which the thread and value are known, but this can only give 

 correct results when the load upon the T is inconsiderable. Portable 

 instruments, which are really carried about and stuck at times out of a 

 window, ought to have the spring to the azimuth-screw such as has 

 been described. 



It is convenient that the clock should be a little slow and have a 

 small losing rate, the corrections for error and rate are then additive : 

 if the west end of the axis be the higher and the deviation to the east 

 of the south, the correction for these errors will also be additive to 

 the observed transits of the greater part of the stars observed. 

 In most cases, the determination of the absolute time at the place is 

 wanted, and this cannot be got without the level or some equivalent 

 which tells how far the instrument swerves from the zenith. But 

 where it is merely required to observe in a meridian, as in observing 

 for a catalogue, it is more expeditious' to change the form of the 



corrections. The two factors 



')- 



COS 



x inclinat. 



8in >- g 

 coa 5 



deviation may be expressed by a correction of this form : m + n tan 8, 

 where m and n are two constants to be determined by observation.* 

 In this case the stars should be observed in zones, and when the 

 sweeps are not near the pole, it is easier to destroy the error of colli- 

 mation by adjustment very nearly than to allow for the error. The 

 secant of decimation varies very slowly, and may be considered as a 

 constant for the whole sweep, within moderate limits, and for a small 

 value of the collimation, which may easily be reduced to 0"1 at once. 

 Suppose an observer to have this purpose : he observes a large set of 

 stars nearly at the same declination, taking care to have as many 

 standard stars aa possible above and below the limits of his sweep, and 

 it is proper to have several with contrary declinations. Now calling 

 the observed times of transit s, d, &c., he forms the following equations 

 with standard stars : 



s + m + n tan S a 



</ + m + n tan 8' = a' 



and so on. From these he composes two equations, one formed of all 

 those in which tan 8 is positive and another in which tan 8 is negative, 

 and which therefore may be represented thus : 



from which n is found 



2 + m + nT A. 

 T + m nT'= A'. 

 (A -2) -(A' -2') 



T + T' 



Substituting this value of n in the mean of the two equations, we 



* If be the inclination as given by the level and x the deviation, then 

 expanding the numerators, the sum of the corrections 



i (cos t. cos 8 + sin #, sin )) +x (sin V . cos > cos . sin >) 



cos 



= ( cos 9+x sin f+(i sin f x cos p) tan t 

 which agrees with the formula given above, putting 



mt+i COB $+? Bin p 



n=t sin p x cos p 



The formula Is easily deduced by drawing a figure and referring the transits to 

 the meridian which cuts the equator at the same point as the circle described 

 by the telescope. 



