minutes past eight it increased from 16 to 43 lbs. in the short space of two or three minutes; the barometer, being at its minimum, suddenly rose about three-hundredths of an inch, and during the heaviest part of the storm it continued to rise at the rate of about one-tenth of an inch an hour. The oscillations in the mercurial column, as will be seen by the diagram, were large and frequent during the storm, one of the most remarkable being immediately after 10 A.M. and nearly coincident with two of the heaviest gusts of wind; the depression in this case amounted to between four and five hundredths of an inch, the rise following the fall so quickly that the clock moved the recording-cylinder only through just sufficient space to cause a double line to be traced by the pencil.

III. "On the Criterion of Resolubility in Integral Numbers of the Indeterminate Equation

f=ax2+a'x12+a"x"2+2bx'x" +2b'xx" +2b"x'x=0."

By H. J. STEPHEN SMITH, M.A., F.R.S., Savilian Professor of Geometry in the University of Oxford. Received January 20, 1864. It is sufficient to consider the case in which ƒ is an indefinite form of a determinant different from zero. We may also suppose that ƒ is primitive, i. e. that the six numbers a, a', a", b, b', b" do not admit of any common divisor. We represent by the greatest common divisor of the minors of the matrix off, by AN2 the determinant off, and by QF the contravariant of f, i. e. the form

(b2-a'a") x2+

... •

A will then be the determinant of F, and Af its contravariant. By , A, and A we denote the quotients obtained by dividing , ▲, and A by the greatest squares contained in them respectively; w is any uneven prime dividing, but not A; 8 is any uneven prime dividing A, but not ; and is any uneven prime dividing both and A, and consequently not dividing A. We may then enunciate the theorem—

"The equation f=0 will or will not be resoluble in integral numbers different from zero according as the equations included in the formulæ

Legendre; the symbols (), (5), (4), (5) are generic characters of ƒ

(see the Memoir of Eisenstein, "Neue Theoreme der höheren Arithmetik," in his Mathematische Abhandlungen,' p. 185, or in Crelle's Journal, vol. xxxv. p. 125).

The theorem includes those of Legendre and Gauss on the resolubility

1864.]

Magnetograph-traces at Kew and Lisbon.

111

of equations of the form ax2+a'x'2+a"x"2=0 (Legendre, Théorie des Nombres, vol. i. p. 47; Gauss, Disq. Arith. arts. 294, 295, & 298). It is equally applicable whether the coefficients and indeterminates of ƒ are real integers, or complex integers of the type p+qi.

It will be observed that if fƒ, ƒ', ƒ" . . . are forms contained in the same genus, the equations f=0, f'=0, ƒ"=0, &c. are either all resoluble or all irresoluble.

IV. "Results of a Comparison of certain traces produced simultaneously by the Self-recording Magnetographs at Kew and at Lisbon; especially of those which record the Magnetic Disturbance of July 15, 1863." By Senhor CAPELLO, of the Lisbon Observatory, and BALFOUR STEWART, M.A., F.R.S. Received January 14, 1864.

The National Portuguese Observatory established at Lisbon in connexion with the Polytechnic School, and under the direction of Senhor da Silveira, has not been slow to recognize the advantage to magnetical science to be derived from the acquisition of self-recording magnetographs. Accordingly that institution being well supported by the Portuguese Government, despatched Senhor Capello, their principal observer(one of the writers of this communication), with instructions to procure in Great Britain a set of selfrecording magnetographs after the pattern of those in use at the Kew Observatory of the British Association.

These instruments were made by Adie of London, and when completed were sent to Kew for inspection and verification, and Senhor Capello resided there for some time in order to become acquainted with the photographic processes. The instruments were then taken to Lisbon, where they arrived about the beginning of last year, and they were forthwith mounted at the Observatory, and were in regular operation by the beginning of July last.

It had been agreed by the writers of this paper that the simultaneous magnetic records of the two observatories at Kew and Lisbon should occasionally be compared together, and the opportunity for such a comparison soon presented itself in an interesting disturbance which commenced on the 15th of July last. The curves were accordingly compared together, and the results are embodied in the present communication.

We shall in the first place compare the Kew curves by themselves, secondly the Lisbon curves in the same manner, and lastly the curves of the two Observatories together.

Comparison of Kew Curves.

The disturbance, as shown by the Kew curves, commenced on July 15th, at 9h 13.5 G.M.T., at which moment the horizontal-force curve recorded an abrupt augmentation of force. The vertical component of the earth's magnetic force was simultaneously augmented, but to a smaller extent; while only a very small movement was visible in the declination curve.

The disturbance, which began in this manner, continued until July 25th, if not longer; but during the period of its action there was not for any of the elements a very great departure from the normal value; probably in this respect the declination was more affected than either of the other com ponents.

While frequently there is an amount of similarity between the different elements as regards disturbances of long period, yet there is often also a want of likeness. If, however, we take the small but rapid changes of force, or peaks and hollows, as has been done by one of the writers of this paper in a previous communication to the Royal Society (Phil. Trans. 1862, page 621), we shall find that a disturbance of this nature which increases or diminishes the westerly declination at the same time increases or diminishes both elements of force. This will be seen more distinctly from the following Table, in which +denotes an increase and a diminution of westerly declination, horizontal, and vertical force respectively, and the proportions are those of the apparent movements of the elements on the photographic paper.

TABLE I.

* Doubtful.

+ Vertical force too small to be accurately measured, but horizontal

force change reckoned =2*0.

From this Table it will be seen that the signs are always alike for the different elements, and also that the small and rapid movements of the horizontal force are double of those of the vertical force-a result in conformity with that already obtained by one of the writers in a previous communication. On the other hand, the declination peaks and hollows do not bear an invariable proportion to those of the horizontal and vertical force, but present the appearance of a daily range, being great in the early morning hours, and small in those of the afternoon. Indeed this is evident by a mere glance at the curves, which, it so happens, present unusual facilities for a comparison of this nature.

Comparison of Lisbon Curves.

1. Declination- and vertical-force curves.-The peaks of the waves, or the elevations in the curve of declination, are always shown in hollows or depressions in the vertical-force curve, and vice versa. We have never seen an instance to the contrary either in the curves under comparison or during the whole time of the operation of these instruments. This curious relation is exhibited in a Plate appended to this communication, from which it will be seen that we have not only a reversal, but also a very nearly constant ratio between the ordinates of the two curves. At Lisbon therefore an increase of westerly declination corresponds to a diminution of vertical force, and vice versá; also an almost constant proportion obtains between the corresponding changes of these two elements.

2. Bifilar and Declination Curves.-July 15. A great disturbance, which at 8 37m Lisbon mean time, or 9h 13m.5 Greenwich mean time, abruptly and suddenly augmented the horizontal force.

The curve of the declination continues nevertheless nearly undisturbed for about 30 minutes after this, and only at 9h 41.5 G.M.T. it commences to descend very slowly.

July 16.-At about 13h 6m G.M.T., a very regularly shaped prominence of some duration occurs in the declination, but is quite invisible in the horizontal force.

July 17.-We see in the bifilar curve half-a-dozen small peaks reproduced in the declination in the same direction, but to a smaller extent.

July 18.-One or two accordant peaks. A large prominence of some duration in the declination at about 17h 56m G.M.T. is reproduced as a slight depression in the horizontal force.

July 19.-A reproduction in the declination of several small peaks of the horizontal force; nevertheless there are others also small which one does not see there, or only reproduced to a small extent. Not much accordance between the great and long-continued elevations and depressions.

July 20. An accordance between the small peaks.

July 21.-The same.

July 22.-The curve is well marked with small peaks.

Coincidence of

several small peaks, but a want of agreement between the more remarkable

peaks. The peaks of the horizontal force more developed than those of the declination.

July 23.-The same appearance of the horizontal-force curves. One remarks on 22nd and 23rd that the small peaks of the declination and horizontal-force are more numerous and more developed in the morning hours.

July 24.-Agreement between the small peaks. A strong disturbance about 10 G.M.T., no agreement between the waves. A well-marked prominence of declination (154) does not alter at all the horizontal-force curve. We derive the following conclusions from the comparison which we have made between the Lisbon curves :

1. The waves and the peaks and hollows of declination are always reproduced at the same instant in the vertical force, but in an opposite direction ; that is to say, that when the north pole of the declination-needle goes to the east, the same pole of the vertical-force magnet is invariably plunged below the horizon, and vice versa. During five months of operation of these instruments there has not been an example of the contrary.

2. The more prominent disturbances of the horizontal force do not in general agree with those of the declination or vertical force either in duration or time.

It is certain that when one of the two elements (bifilar or declination) is disturbed, the other is also; and sometimes one appears to see even for several periods of one of the curves, an imitation of the general march of the other; but when this is examined a little more minutely, and rigorous measures are attempted, one easily perceives that the phases do not arrive at the same time, but sometimes later and sometimes earlier, without any fixed rule.

In the same curve one generally sees contradictions of this kind. Nevertheless it is certain that the agreement in direction and time is more complete when the elevations or depressions are of shorter duration.

3. The small peaks and hollows are generally simultaneous for the three curves. The direction of these is the same for the horizontal force and declination, while that for the vertical force is opposite.

The ratio in size of the peaks and hollows is generally variable between the horizontal force and the declination, while it is always constant between the latter and the vertical force.

Our next deduction requires a preliminary remark. It has been shown by General Sabine, that if the disturbances of declination at various places be each divided into two categories, easterly and westerly, these obey different laws of daily variation, this difference not being the same for all stations.

This would seem to indicate that for every station there are at least two simultaneous disturbing forces acting independently, and superposed upon one another.

This interesting conclusion, derived by General Sabine, appears to be