for the mean diurnal inequalities of winter, summer, and the whole year.

The a, b coefficients for the 'year' are the arithmetic means of those for 'winter' and 'summer'; but as a check on the accuracy of the work they were calculated independently.

Variation to the west in the declination is regarded as positive.

The only change required in Table V. if time were measured from noon, instead of as throughout the present paper from midnight, would be the alteration of the signs of the entries referring to a, b, ag, and b3.

It should be explained that the calculations on which Table V. is based proceeded in every case to at least one decimal place, usually two, beyond that shown.

The table shows that in the case both of the declination and the horizontal force, the terms whose period is twenty-four hours are very much greater in summer than in winter, and the same phenomenon shows itself in the terms in the declination whose period is twelve hours. On the other hand the terms in the declination whose period is six hours appear relatively much larger in winter than in summer. It would hardly be prudent to attach too much weight to this last conclusion, in view of the extreme smallness of the quantities involved; but it is in accordance with Tables XV. and XVI. of the 'Greenwich Magnetical and Meteorological Observations' for the two years 1890 and 1891. A comparison of the other features common to these two Greenwich tables and to Table V. will be found of interest. The Greenwich tables, it should be noticed, are not confined to 'quiet' days, and in treating the horizontal force they take as unit the (1/100000) of this force at Greenwich, or, say, 10-8 × 182 C.G.S. unit.

§ 10. The number of degrees in the angles 2nT/24 is easily obtained from Table V. by means of the formula (3). Instead of giving these angles explicitly I have preferred to give the times 7, answering to the earliest maxima in the day of the Westerly declination and the horizontal force. These times are included in Table VI., along with the earliest times in the day when the several terms of the types appearing in (1) vanish and have their minima. The interval between successive maxima or successive minima is double that between successive zeropoints, and equals the periodic time, i.e., is 24, 12, 8, or 6 hours, as the case may be.

With one important exception-that of the harmonic term in the declination whose period is twenty-four hours the first maximum of every term appears earlier in the day in summer than in winter. The difference between the corresponding times in winter and summer is

conspicuously greater in the case of the horizontal force than in that of the declination.

There is, it will be noticed, a close coincidence between the time, 9 hr. 38 min., of the minimum in summer of the horizontal force term whose period is twenty-four hours and the time, 9 hr. 31 min., of the first zero in summer of the corresponding declination term. The fact, however, that no such coincidence presents itself between the corresponding times in winter would suggest the phenomenon was in part at least accidental. It can hardly be connected with the fact-shown conspicuously by Tables III. and IV., or still better by curves 1 to 10-that the time of the principal minimum of the total horizontal force inequality, from 10 to 11 A.M., is nearly coincident with a time at which the total declination inequality vanishes. Papers dealing with a theoretical connection of the sort, by Professor A. Schuster and Mr. C. Chambers, will be found in the 'Phil. Mag.' for April and May 1886, and in 'Proc. Lit. and Phil. Soc.,' Manchester, Session 1886-87, pp. 23-46.

Resultant of Cyclic Horizontal Disturbing Forces producing the
Diurnal Inequality.

§ 11. The importance attaching to variations in declination suggests most naturally the resolution of the horizontal disturbing force, to which the diurnal inequalities just considered are due, into components ¿X and Y respectively in and perpendicular to the magnetic meridian. The preceding results show that, H denoting as usual the total horizontal force, ¿X/H and Y/H are at most small quantities of the order 1/500. Thus to a very close approximation we have

where H and D are the inequalities of horizontal force and declination, while H is the mean value of the horizontal force for the time under consideration.

where ¿D is supposed to be given in minutes of arc.

(5)

Resolution in and perpendicular to the magnetic meridian is somewhat inconvenient when the general features of terrestrial magnetism are being considered, owing to the complicated relationships between the magnetic meridians at different stations. Even in dealing with the phenomena extending over a long interval of time at a single station, the reference to a set of axes whose position is continually altering has its disadvantages. When, however, the components of a force along two rectangular axes are known, the components along any other pair in the same plane are easily deduced when the inclination of the old to the new axes is given. It will thus suffice to state that for the epoch 1890-94 the mean magnetic meridian at Kew lay at 17° 36'2 to the west of the astronomical meridian.

§ 12. Instead of giving explicitly the components in and perpendicular to the astronomical meridian I propose to consider the magnitude and direction of the resultant horizontal disturbing force itself. Denoting its intensity by p, its inclination to the east of magnetic north by, we have to a sufficiently close degree of approximation

=√(@X)2+(oY)2=&H sec

tan-(-dY/EX)=tan1 (¿D. H/H)

Remembering that the inclination of p to the astronomical north is -(17° 36′2), it would be found a simple matter to deduce the components of the disturbing force along and perpendicular to the astronomical meridian from the calculated values of and .

The following table, VII., gives the values of and for the mean of December and January, or midwinter, the mean of June and July, or midsummer, and the mean for the year at each hour. The greatest and least values of p in each case are in heavy type.

In comparing Tables IV. and VII. the reader will require to notice that the former refers to 106 cH, the latter to 107. Table VII. was got out from arithmetical means proceeding to two places beyond those retained in Table IV., and an extra figure was retained in, so that the arithmetical accuracy of any resolution of p into orthogonal coordinates should be as trustworthy as its previous resolution into X and Y, i.e., into ¿H and H.D.

The angle 360° is to be added to the values of under the hour 23 for 'midsummer,' and for hours subsequent to 18 in 'midwinter' and ' ' year,' when it is desired to make out the true increase of relative to a previous hour.

Ρ

In the case of the yearly mean the vector has a continuously progressive rotation from east to west through south, like the sun as seen from the earth. This is the general character of the rotation at any season; but at midwinter there is an unmistakable retrograde motion in the early morning for several hours, and at midsummer there is at least a suspicion of a retrograde motion from 8 to 10 P.M. From 10 A.M. to