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 ment. Form four groups, two in each set, those in which x has the largest, and those in which x has the smallest coefficients, dividing each group by the number of its component parts, so as to leave e with unity for its coefficient. Call these equations 1, 2, 3, and 4. Sub tracting (4) from (1), we shall eliminate s, c will have a + coefficient exceeding 2, and x in the most difficult case, that is, when the observer can only 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 two equations c can be determined pretty accurately. Substitute this value in equations (1) (2) (3) snd (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, e is eliminated and x determined with tolerable accuracy. Finally, the substitution of these values of c and x in the original equations will give a satisfactory clock-error if the observations are good and pretty 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 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 reason to fear any alteration in the position of the stand or the VS 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 coefficients of his unlinarn quantities, a discretion which some astrono mars cannot or will not use.
It is always desirable that the value of the three transit corrections should be small (indeed the formuke 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 1' 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 bo 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 x inclinat. + x cos 8 cosh deviation may be expressed by a correction of this form : +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 declination 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 largo set of stars nearly at the same declination, taking care to have as many standard stars as 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, a', &c., he forms the following equations with standard stars :— a+nt+ntan8=a + m + n tan 8' = a' and so on. From these ho composes two equations, one formed of all those in which tan 8 is positive and another in which tan a is negative, and which therefore may be represented thus : ?+m+nT=A. from which n is found= (A —1)— (A' — T + T' Substituting this value of n in the mean of the two equations, we * If f be the Inclination as given by the level and x the deviation, then expanding the numerators, the sum of the corrections (cos p. cos a +sin 9, sin a) +x (stn 9. cell a —cos 9, sin 3) cos a =1 cos p+x sin 9-1-(i sin p—z cos ;) tan a which agrees with the formula given above, putting 01=s+i cos p+x sin 9 n=i sin Co. p The formula Is easily deduced by drawing a figure and referring the transits to the meridian whloh cuts the equator at the same point es the circle described by the telescope.