We have the following expressions, in determinant notation*::

[de. 3] = [waa][wbb][wcc] [wde] ÷ [waa] [wbb][wcc],

[ee. 3] = [[waa][wbb][wcc][wee] ÷ [[waa][wbb][wcc]|.

Or, since e, the coefficient of AT, in the equations of condition, =1—

[de. 3]=[waa] [wbb]]wcc] [wd]] [[waa] [wbb][wcc],

[ee. 3] = [waa][wbb][wcc][w]] ÷|[waa][wbb][wcc].

The only coefficient affected by a change of epoch for AT, is d, the coefficient of the rate. The denominator of [de. 3] is independent of d, and, as it is not infinite, to make [de. 3]=o its numerator must become zero. This numerator expressed in the fuller determinant form is

Let us suppose that we have formed and solved our normal equations with the epoch T for ATO, and let us change the epoch to T'T+t'. The coefficient of the rate is the time of observation minus the epoch of ATo. We shall have the following relations between the two sets of coefficients, d representing in both sets the coefficient of the rate for the epoch T.

* These expressions, as well as those for the other coefficients occurring in the solution of the normal equations (Chauvenet, ii. pp. 531 and 532), can be found by expressing each as a determinant of the second order; e.g.

substituting in each the determinant expressions for the coefficients of lower order, and applying the theorem

The first member is the determinant whose element in the kth row and

Zth column is

a11
a(k+1)1
; k, l = 1, 2, 3 .
a1(+1) α(k+1)(+1)

....

(m−1).

Let us substitute the values for the epoch T' in the numerator of [de. 3].

We have

No. of [de. 3] for epoch T'=

[waa][wab][wac][wad] - [wa]. t']
[wab][wbb][wbc] [wbd]-[wb]. t'
[wac] [wbc] [wee] [wed]-[we]. t'
[wa] [wb] [we] [wd]-[w].t'

Splitting up this determinant into two and factoring out t in the second, we have, for the epoch T'—

No. of [de. 3]=[waa][wbb][wcc][wd]|-|[waa] [wbb][wcc][w]. t'.

Putting this equal to zero and noticing that the first term and the factor of t in the second term are proportional to [de . 3] for the epoch T and [ee. 3] respectively (4), we find for t' the interval from the epoch assumed for AT, to that which would give the maximum coefficient for AT, in the final equation, or to the epoch at which the clock correction has the minimum probable error,

The value found for t' is the quotient of the same coefficients, after the equations have been freed from all the unknowns except AT。 and p, of which it would have been the quotient, had there been only these two unknowns in the original equations, namely, of the coefficient of AT。 in the normal equation in p by that in its own normal equation. It will, in general, differ from [wde] [wee] the mean of the times of observation.

In the example given below the square brackets are used also to indicate the number whose logarithm they enclose.

Summary.

The epoch of the clock correction of maximum weight, or minimum probable error, is not, in general, the mean of the times of observation when, besides the constant clock correction and the clock rate, instrumental constants are also determined from the observations, (5).

If these quantities are found by a least squares solution, this epoch is before or after the epoch assumed for the constant clock correction by an interval which is the quotient of the coefficient of the constant clock correction in the normal equation for the rate by its coefficient in its own normal equation, after the elimination of the other unknowns, (5).

If we count from this epoch, the probable error of the clock correction at any other time is what it would be if the constant correction and the rate were independently observed quantities

that is, its square is the sum of the square of the probable error at this epoch and the product of the square of the probable error of the rate into the square of the interval from this epoch, (1). Hence the square of the probable error of the clock correction at this epoch is equal to the square of the probable error of the clock correction at the assumed epoch minus the product of the square of the probable error of the rate into the square of the interval between the two epochs.

Example.

Number of stars observed, 20. Number of quantities determined, 5.
Putting v here for the residuals, [wvv]=0°0251.

For an observation of weight I, p2=0000761 = [6.8815].

Mean of the times of observation = 8h.98. Assumed epoch of AT=91.0. [ee. 4] 328, (PAT)2= [6·8815]÷[0·5159]=[6.3656]=0000232. (pp) [6.8815]÷[0·6474] = [62341]=0000171.

[dd. 4]=4'44,

+575 [07597], [ce. 3]= +6·85 = [0·8357], t'=[9.9240]=oh.84.

Epoch of maximum coefficient = 9h.84.

At time T, pAT ± √0·000111+0000171. (T-9h.84)2.

Georgetown College Observatory,
Georgetown, D.C., U.S.A.

On the Photographic Magnitude of Nova Auriga, as Determined at the Royal Observatory, Greenwich. II. By W. H. M. Christie, M.A., F.R.S., Astronomer Royal.

In the Monthly Notices for last March (vol. lii. p. 357) results for the photographic magnitude of Nova Auriga were given as determined up to March 9. Since then photographs have been taken up to April 1, when the photographic magnitude had fallen below 14, and again at the end of August and beginning of September, when the star had brightened. Further measures were also made of the earlier photographs, modifying slightly the results given in the previous paper. The measures were made and reduced in the manner explained in that paper, the magnitude of the Nova being inferred by comparison with four Argelander stars of 8-9 magnitude by means of the formula

m=25 (log t-0.97 √) + const.

given in the Monthly Notices, vol. lii. p. 146, the measures of diameter of the four comparison stars being used to determine

the value of the constant for each plate. Their magnitudes taken from Argelander are: B.D. +30°·944, 82; +30°°949, 82; +30°938, 8'7; +30°·913, 87; and it is assumed that the mean magnitude of the four stars is 8'45.

The

The results are given in the accompanying Table I. initials A.E., E.R., A.R., W.C., C. are those of Miss Everett, Miss Rix, Miss Russell, Mr. Christie, and Mr. Criswick respectively.

TABLE I.

Photographic Magnitude of Nova Aurige from Comparisons with
Four Stars, 8-9 Mag.

The number of images of the Nova measured on each day is indicated by the suffix, the exposures ranging generally from 15s to 12m, as shown in the previous paper. This scale of exposures was, however, to some extent modified when the Nova got

too faint to be shown with the short exposures, and the following is a list of the exposures from March 19: March 19 and 24, 12m, 4m, 1m; March 28, 12m, 6m; March 30, 20m, 10m, 30s, 159; April 1, 30m, 15m, 30, 15s; August 30 and September 3, 12m, 4m, Im; September 5, 12m, 5m, 4m, Im; September 6, 81m, 4m, 21m, Im, 30s. On March 30 the Nova was shown with an exposure of 10m, on April 1 with an exposure of 30m, and on September 6 with an exposure of Im.

4

On March 15 and 18 two sets of exposures were given, the first with the Nova, and the second with the 5'7 mag. star B.D. +30°.898, at the centre of the plate, the Nova in the second set being about 60' from that centre. These plates were taken to test whether distance from the centre would appreciably affect the determination of photographic magnitude.

The following are the values of photographic magnitude of BD + 30° 898 deduced in the two cases :

In connection with these measures it may be of interest to give the value of the constant deduced from each plate (the meaning of which has been explained in the previous paper, p. 364), Table II., and the results obtained for the photographic magnitudes of the four comparison stars referred to the mean of the four, which is assumed from Argelander to be 8:45, Tables III. and IV.

In explanation of the small values of the constant in some cases, it is to be remarked that several of the nights on which the photographs were taken were cloudy or hazy. This was the case on March 7, 9, 10, 14, and 15.