The partial sums of the columns containing A as a factor and the total sums of the other columns are the coefficients of the normal equations.

Hence applying them as in the formula above given we have the normal equations :

(4)

+54.196 a, -72 877b89-918 c-5.470 (5)—22.263a2+7·808a2+6·699a-72.877a,+186-2306+222-759c+9.544

+ 0.248 +12-562

(6)—26.344a,+8·757a2+7·404a3+89-918a,+222-7596+266-935c+11·8568=+14·524

(7) 2.092 a,-0·822 a2-0·479 az-5470 a, +9.544 b+11-856 c+16·440 &r=+ 0.045 The solution of these is as follows:

=

Here in line I are entered the coefficients of normal equation (5), and of equation (1) in line II. Line III = — II ÷÷ 16·004 × 22.263, and line IV = line I+ line III. Line V equation (2), from which VI is got by division by 5.499 and multiplication by 7.808, changing the signs. VII is the sum of IV and VI.

=

By repetition of this process we obtain at last line XIII, which is equivalent to the equation :

37.2243 642-8718 c + 1.0856 +10.0769.

=

By an exactly similar process with equation (6) we get an equation:
42.8718649 5045 c + 1·3534 dt = +11.6495

And by treatment of equation (7) we get

+1.0856 b + 1·3534 c + 15.4458 d

=- -0.0060

From these three equations c is eliminated by a process similar to the above, giving us two equations:

we

+ 0.09657 b

008648 δτ

0.08648b15.40881 d

By another elimination, we get

0.01179 0.32451

+15.33137 or =— 0.33507
Whence or

—— Os.022

And its weight is 15.331

By substituting the value of or in one of the equations involving b and d only, get b =— 0-142.

So, by successive substitutions, we get c = 0·3586, a, = + 0·304, a ̧ = -0.215. a =- – 0·394, a =

= + 0·407.

Now, correcting the residuals d of each star with the quantities Aa, Bb, Cc, and ST found thus, we get the true residuals as follows:

+ ⚫062 + ·008

+ .079

The sum of the squares of these residuals is ⚫164496.

The number of observations is 17 and the number of unknowns = 7.
Hence the mean error of an observation is:-

Dividing this by the square root of the weight of St gives the probable error of the deduced S as ± 0.022.

Hence, finally the chronometer correction at the time 2h 00m is 4h 02m 598 204 08 022 4h 02m 59, 226 with probable error = 022. Adding the value of b found above to its value assumed at the beginning of the calculation we find for the inequality of pivots the value 1.02. This value is found

to have a large probable error, and to vary a considerable fraction of itself in the results of the observations of different nights. So also does the collimation, it increasing when b decreases, and vice versa.

This indicates the weak point of this method of determining these instrumental constants b and c from the star observations. It is this, that when the instrument is reversed in order to reverse the effect of the error of collimation, the effect of the inequality, b, is also reversed. And their coefficients B and C being always nearly equal to one another, we are in reality determining each from an indirect observation of a very small fraction of it. Hence the result is inaccurate.

The normal equations numbers 5 and 6 are the equations in which b and c are the principal terms respectively. Now in (5) the coefficient of b is B and of c, BC, and in (6) the coefficient of b is BC and of c, C2.

Now BC cos (4-8) whence B and C have always the same algebraic sign, and BC is always positive.

Also C is always greater than B, hence [B], [BC], and [C] are always, whatever stars be selected, of the same sign, and in ascending order of magnitude, although not very different in magnitude.

Hence the difficulty spoken of above, of separating b from c. A slight change in the residuals d caused by the ordinary accidental errors of observation will then have a great effect on the deduced values of b and c. But they change in opposite directions, and to some degree proportionately to the B and C of the stars, so that they have not a great effect upon & which is the important matter in these obser

vations.

To determine these constants accurately by astronomical observation it would be necessary to take some of the stars such that B and C would be of different signs. This requires that cos (6-6) be negative or the stars taken by reflection as from the surface of mercury. But the same result can be arrived at in an easier manner, as follows using the fixed threads of the instrument instead of the star,—that is, to look vertically downwards through the telescope into a mercury trough and measure the distance of the middle thread from its reflected image with the movable micrometer thread, and to perform a similar operation in looking upwards to the reflection of the threads from the surface of water contained in a vessel with a transparent bottom placed vertically over the object glass.

These two observations give, the one the sum of the pivot inequality and the collimation error, and the other their difference, whence each can be determined. This is not an easy observation, on account of the difficulty of throwing a proper amount of light on the threads. My determinations by this method are not yet complete. If these constants are well determined, the observations can then be worked out in sets of five stars each, so as to reduce the corrections to be applied for rate, by reducing each set to its middle instant, the reduction of the inaccuracy due to a wrong assumed rate being an important matter with a chronometer, such as the one I used, having a large rate coefficient, at low temperatures.

The errors of transit observations are due chifly to errors of bisection of the star, or of estimating the exact moment when the star is on the thread, tremors of the instrument, irregular and lateral refraction, irregular expansions of parts of the instrument from the heat of the lamps, errors in the tabulated places of the star used &c.

The errors in the tabulated star places were avoided by using stars from one catalogue (Berliner Jahrbuch), and by both observers using the same stars; so that any errors in the right ascensions will affect the local times deduced for each place by equal quantities, and this error will disappear from their difference, i.e., the difference of longitude.

As far as an error in bisection is constant and in the same direction, and peculiar to the observer, it is found and allowed for as a personal equation as will be shown below. Accidental errors of bisection, and the tremors of the instrument, &c., are of the same kind and may be considered together. The amount of this error may be estimated by comparing the equatorial thread intervals as obtained from the different 14-21**

observations with the true equatorial intervals of the threads as determined from careful observations of slow moving stars, or otherwise. The various residuals give, by the use of the least square formula, the probable error of a bisection.

I have so computed all my observations on 30th September, 1st, 2nd, 3rd, 4th, 5th, 9th, 10th and 12th October, and find the probable error of the transit of an equatorial star over one thread to be± 0.049, or 08 023 for the mean of five threads. This result is obtained from 436 observations. The probable error of a bisection of any star may probably be taken to be 0.023 sec. ò or 0.023 C. The probable error from this cause of the mean of all the observations of the 4th October, given above, will then be 023 [PC which taking all the "time" stars, or those south of the

V1644

zenith gives a probable error of about 08.02. This result agreeing closely with the probable error found above from the observations directly indicates that the discrepancies of these observations are mainly due to accidental error of bisection. I may add that all my transits were recorded by means of an electric key on the tape of my Morse register.

The errors of the chronometers having been thus determined, we now come to the means by which the two chronometers at distant places were compared. For this purpose each observer was provided with "break circuit chronometer, a Morse tape register and a "switch board" with the necessary relays and keys for transmitting the electric signals. In the chronometer, a small toothed wheel, within the case, raising a spring, at the end of every even second, breaks an electric current passing through the chronometer.

On the switch board were, a small relay worked by the chronometer local current, a sounder with relay, and another relay for transmitting the signals from the main line to the local circuit which worked the register. These last two relays had each a resistance of about 200 ohms and were in the circuit of the telegraph line, which by the courtesy of the Canadian Pacific Railway officials was brought into our observatories. There were also on the board a "break circuit" and a "make circuit" key in the line circuit and a number of stops and switches for changing the connec tions as required. For exchange of signals the line circuit was brought through the armature points of the chronometer relay, so that at every two seconds break of the chronometer, the armature flying back broke the line circuit, which break by means of one of the large relays (which I call for distinctness the "signal relay," the other one, the "speaking relay" being used only in connection with the sounder for con. versation over the line), breaks the local circuit of the register, and so lifting the tracing point from the tape, leaves a break in the otherwise continuous line on the paper.

Any other break in the line is also by the signal relay transferred to this tape. So that if both chronometers are working on the line at the same time, each tape receives the breaks of both.

The tape runs off the reel at the rate of about 23 inches in two seconds, so that a marked tape shows a continuous line broken by two regular series of short breaks, the successive breaks of each series being about 23 inches apart.

The ratio of the distance between a break of one series and the next following break of the other series to half the distance between successive breaks of the first series, gives the fraction of a second between the even seconds of the two chronometers. The beginning of the break is the point measured from in all cases.

The breaks of the two chronometers are distinguished by putting one chronometer into circuit eight or ten seconds before the other, and the even minute is dis tinguished by its being at the end of a double length of unbroken line, the chronometers omitting the usual break at the 58 seconds.

The errors peculiar to this apparatus are as follows:

The mechanism of the chronometer may be imperfect, so that the breaking of the circuit does not occur at exactly the moment when the second hand of the chronometer indicates the second.