paper. These sheets are therefore only changed every second day. This change is made a little after 10 A.M., and the time occupied in making it is about ten minutes, while that occupied in pushing forward the dots is only about three minutes. There is thus every day a loss of ten and of three minutes alternately, so that the curves never record precisely the whole of the twenty-four hours, but generally something less by a few minutes. The precise moment (Kew mean time) of stopping the pendulum and of setting it going again is noted, so that the length of time for which any curve is a record is known and is attached to the curve in writing. (See curves appended to this Report, Plate 5.)

The instrument for tabulating from the curves is represented in Plate 3. fig. 3 A: ab is a time-scale commencing and ending with 22h. This scale is moveable round a as a centre, and the centre a is also moveable in a horizontal direction. Part of the instrument, df g, is moveable in a vertical direction by means of h, the head of a pinion which works into the rack i; d serves as a vernier for the scale e. The piece c d e f g is moveable in a horizontal direction by means of a slide which fits into the slot k l; ƒ and g are two tubes through which the eye looks at lines on a piece of glass (exhibited separately at full size in fig. 3a). These are two sets of double lines which are etched on glass, the sets being exactly two inches apart. The distance between the tubes ƒ and g is also two inches, so that when the upper pair of lines is placed under f, the lower pair is under g. The glass is firmly attached in this position to the moveable piece dfg, so that the double lines remain exactly under the tubes in whatever manner dfg is moved. The breadth between the two lines (which together constitute a double line) on the piece of glass is a little greater than the breadth of the curve or zeroline on the photographic paper.

In order to measure the distance between the curve and zero-line, the photographic paper is set between two pieces of plate-glass, and so adjusted, that when the tube g is set over the zero-line, it may continue to be approximately over it in any part of its horizontal range.

Suppose now that c d e f g is at the extreme left, the vertical line of the piece of glass lying along the commencement of the curve and that of the zero-line. Set the time-scale ab so that the edge of the index e may touch that hour on the time-scale which corresponds to the commencement of the curve. Adjust the vertical height of b, the extremity of the time-scale, so that when c d e f g is carried to the other or right-hand extremity of the curve, the index e may touch that division of the time-scale which corresponds to the termination of the curve. Were the same length of base-line always to denote the same space of time, there would be no need of altering the inclination of a b; but the rate of the clock may vary a little, or the paper inay fit more or less loosely to the cylinder, so that an inch of the base-line will not always denote precisely the same space of time. Having thus adjusted the time-scale, in order to find the distance between the base-line and the curve for any hour, set the index e to the required time, move the pinion head h until the upper pair of etched lines at ƒ are over the curve-line, and read off the height on the scale e by means of the vernier d. Next move the pinion head h until the lower pair of etched lines at g are over the baseline, and read off by means of the vernier as before. The difference between the readings for the curve and the base-line plus two inches, gives the distance between these lines.

In case any shifting should take place, it is best to read the curve and its corresponding base-line consecutively, instead of reading first a number o points of the curve together, and then the corresponding points of the base

line together also. Occasionally the presence of iron for a short time may cause an abrupt rise and fall of small size in the curve, the one motion being due to the approach of the iron, and the other to its removal. These must be taken into account in tabulating from the curves. An instance of this occurs in the curves appended to this Report.

Section V. IMPROVEMENTS IN THE CONSTRUCTION OF A SET OF SelfRECORDING MAGNETOGRAPHS SINCE MADE.

Magnetographs very similar to those here described have been lately set up in a house constructed to receive them about 70 yards from the Kew Observatory.

The following improvements were made in their construction :—

1. Instead of one large glass shade standing upon the marble slab, each magnetograph has a gun-metal cylinder, which stands upon the slab, and is surmounted by a glass shade of comparatively small size. An opening is cut in the side of the cylinder, in which there is inserted a piece of perfectly plane glass; this glass covers that space which in the old arrangement would have been occupied by the two round holes already described. The lens is apart from the cylinder, and has an adjustment to admit of its distance from the mirror being altered if necessary.

This arrangement permits the shades to be removed without disturbing the lenses. It also renders the working of the instrument less liable to interruption in case of any accident happening to the shade.

There is also a tube inserted through the marble, which may be connected with an air-pump and the interior of the cylinder and shade exhausted, if this be thought necessary.

2. The second improvement consists in having reading telescopes with ivory or other scales mounted on pillars, and so placed that the light from the divisions of the scale falling upon the moveable mirror attached to the magnet is reflected into the telescope. In consequence of this, the motion of the mirror will cause an apparent motion of the scale in the field of view of the telescope. The position of the magnet will therefore be known by observing what division of the scale is in contact with the vertical wire of the telescope.

We may thus combine the photographic record with eye observations. The advantage of the latter is that we see what is taking place at the very moment of its occurrence, whereas we only obtain the photographic record a couple of days after the changes to which it relates have happened.

Should a disturbance take place, we are thus not only made aware of it at the time of its occurrence, but we may, by having a telescope scale of greater range than the recording cylinder, obtain eye observations, when owing to excessive disturbance the dot of light has altogether left the sensitive paper.

Report on the Theory of Numbers.—Part I.

By H. J. STEPHEN SMITH, M.A., Fellow of Balliol College, Oxford. 1. THE Disquisitiones Arithmetica' of Karl Friedrich Gauss (Lipsiæ, 1801) and the Théorie des Nombres' of Adrien Marie Legendre (Paris, 1830, ed. 3) are still the classical works on the Theory of Numbers. Nevertheless, the actual state of this part of mathematical analysis is but

imperfectly represented in those celebrated treatises. The arithmetical memoirs of Gauss himself, subsequent to the publication of the 'Disquisitiones Arithmetica;' those of Cauchy, Jacobi, Lejeune Dirichlet, Eisenstein, Poinsot, and, among still living mathematicians, of MM. Kummer, Kronecker, and Hermite, have served to simplify as well as to extend the science. From the labours of these and other eminent writers, the Theory of Numbers has acquired a great and increasing claim to the attention of mathematicians. It is equally remarkable for the number and importance of its results, for the precision and rigorousness of its demonstrations, for the variety of its methods, for the intimate relations between truths apparently isolated which it sometimes discloses, and for the numerous applications of which it is susceptible in other parts of analysis. "The higher arithmetic," observes Gauss*, confessedly the great master of the science, "presents us with an inexhaustible store of interesting truths,-of truths, too, which are not isolated, but stand in a close internal connexion, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed."

2. It is the object of the present report to exhibit an outline of the results of these later investigations, and to trace (so far as is possible) their connexion with one another and with earlier researches. An attempt will also occasionally be made to point out the lacunae which still exist in the arithmetical theories that come before us; and to indicate those regions of inquiry in which there seems most hope of accessions to our present knowledge. In order, however, to render this report intelligible to persons who have not occupied themselves specially with the Theory of Numbers, it will be occasionally necessary to introduce a brief and summary indication of principles and results which are to be found in the works of Gauss and Legendre. It is hardly necessary to add that we must confine ourselves to what we may term the great highways of the science; and that we must wholly pass by many outlying researches of great interest and importance, as we propose rather to exhibit in a clear light the most fundainental and indispensable theories, than to embarrass the treatment of a subject, already sufficiently complex, with a multitude of details, which, however important in themselves, are not essential to the comprehension of the whole.

3. There are two principal branches of the higher arithmetic :-the Theory of Congruences, and the Theory of Homogeneous Forms. The first of these theories relates to the solution of indeterminate equations, of the form an x" +α-12"-1+....+a1x+a2=Py,

'+

in which a a... a, a, and P are given integral numbers, and x and y are numbers which it is required to determine. The second relates to the solution of indeterminate equations of the form

F(x, x,...x)=M,

in which M denotes a given integral number, and F a homogeneous function.

* Preface to Eisenstein's Mathematische Abhandlungen,' Berlin, 1847.

of any order with integral coefficients. In this general point of view, these two theories are hardly more distinct from one another than are in algebra the two theories to which they respectively correspond, the Theory of Equations, and that of Homogeneous Functions; and it might, at first sight, appear as if there was not sufficient foundation for the distinction. But, in the present state of our knowledge, the methods applicable to, and the researches suggested by these two problems, are sufficiently distinct to justify their separation from one another. We shall therefore classify the researches we have to consider here under these two heads; those miscellaneous investigations, which do not properly come under either of them, we shall place in a third division by themselves.

(A) Theory of Congruences.

4. Definition of a Congruence.-If the difference between A and B be divisible by a number P, A is said to be congruous to B for the modulus P; so that, in particular, if A be divisible by P, A is congruous to zero for the modulus P. The symbolic expressions of these congruences are respectively A=B, mod P, A=0, mod P.

Thus 72, mod 5; 13=−3, mod 8.

It will be seen that the definition of a congruence involves only one of the most elementary arithmetical conceptions,-that of the divisibility of one number by another. But it expresses that conception in a form so suggestive of analogies with other parts of analysis, so easily available in calculation, and so fertile in new results, that its introduction into arithmetic (by Gauss) has proved a most important contribution to the progress of the science. It will be at once evident, from the definition, that congruences possess many of the properties of equations. Thus, congruences in which the modulus is the same may be added to one another; a congruence may be multiplied by any number; each side of it may be raised to any power whatever, and even may be divided by any number prime to the modulus.

5. Solution of a Congruence.-If (x) denote a rational and integral function of a with integral coefficients (we shall, throughout this report, attach this meaning to the functional symbols F, f, o, &c., except when the contrary is expressly stated); the congruence (x)=0, mod P, is said to be solved, when all the integral values of x are assigned which make the left hand number of the congruence divisible by P; i. e. which satisfy the indeterminate equation p(x)=Py. It is evident that if x=a be a solution of the congruence p(x)=0, every number included in the formula x=a+μP is also a solution of the congruence. But the solutions included in that formula are all congruous to one another and to a. It is proper, therefore, to consider all these congruous solutions as identical, and in speaking of the number of solutions of a congruence to understand the number of sets of incongruous solutions of which it is susceptible. To assign, by a direct method, all the solutions of which a proposed congruence is capable, is the general problem which, in the Theory of Numbers, corresponds to the problem of the solution of numerical equations in ordinary algebra. But the solution of the arithmetical problem is attended with even greater difficulties than that of the algebraical one; and the attention of geometers has been turned with more success to the improvement of the indirect or tentative methods of solution, and to the discovery of criteria of possibility or impossibility for congruential formulæ, than to their direct solution. It is to be observed that, by virtue of a remark already made, the tentative

solution of a congruence involves no theoretical difficulty. For if x=a be a solution, every number included in the formula x=a+μP is also a solution, and among these numbers there is always one, and only one, comprised within the limits O and P-1 inclusively. By substituting, therefore, for x all numbers in succession less than the modulus, and rejecting those which do not satisfy the congruence, we shall obtain its complete solution. But the interminable labour attending this operation, notwithstanding all the abbreviations in it suggested by the Calculus of Finite Differences, renders its application impossible, except when the modulus is a low number.

6. Systems of Residues. The set of numbers 0, 1, 2.... P-1 (or any set of P numbers respectively congruous for the modulus P to those numbers) is termed a complete system of residues for the modulus P. By a system of residues prime to P, we are to understand a complete system, from which every residue has been omitted which has any common divisor with P. Thus 1, 5, 7, 11, or 1, 5, −5, −1, are the terms of a system of residues prime to 12. The word Residue is employed instead of Remainder, because the word Remainder would suggest the idea of a positive number less than the modulus or divisor; whereas it is frequently convenient to consider residues differing from those positive remainders by any multiples of the modulus whatever.

7. Linear Congruences.-The general form of a linear congruence is ax+b=0, mod P; a, b, and P denoting given numbers, and x a number to be determined.

The theory of these congruences may be considered to be complete, both as regards the determination of the solutions or roots themselves and of their number. If a be prime to the modulus, there is always one solution, and one only; if a have a common divisor with the modulus which does not also divide b, the congruence is irresoluble; if ♪ be the greatest common divisor of a and P, and if & also divide b, the congruence has d solutions. In every case when the congruence is resoluble, the direct deterinination of its roots may be made to depend on the solution of a congruence of the form ax=1, mod P, in which a is prime to P. This congruence coincides with the indeterminate equation ax=1+Py, methods for the solution of which were known to the ancient Indian geometers*, and have been given in Europe by Bachet de Meziriac + Euler, and Lagrange §. The methods of these writers ultimately depend on the conversion of a vulgar fraction into a continued fraction, and in one form or another have passed into every book on algebra. Nor would it have been proper to allude to them here, were it not that they serve to supply us with a clear conception of what we have a right to expect in the solution of an arithmetical problem. In such problems, we cannot expect to express the quæsita as (discontinuous) analytical functions of the data. Such expressions may indeed, in many cases, be obtained (by the use of the roots of unity or by other methods); but the results of the kind which have hitherto been given, though sometimes of use in calculation, may be said, with few exceptions, to conceal rather than to express the real connexion between the * See the Arithmetic of Bhascara, cap. xii., and the Algebra of Brahmegupta, cap. i. in Mr. Colebrooke's translation, London, 1817.

† Problèmes plaisans et délectables, qui se font par les nombres. Seconde édition. Par Claude Gaspar Bachet, Sieur de Meziriac, Lyon, 1624. (See props. xv. to xxv.)

Comment. Acad. Petropol. tom. vii. p. 46, or in the Collection of Euler's Arithmetical Memoirs (L. Euleri Commentationes Arithmetica Collectæ, Petropoli, 1849), vol. i. p. 2; and in his Elements of Algebra, part ii. cap. 1.

§ Sur la Résolution des Problèmes Indéterminés du seconde degré. Hist. de l'Acad. de Berlin, 1767, p. 165. (See Arts 7, 8, and 29 of the Memoir.) Also in the Additions to Euler's Algebra, sects. i. and iii. (Lyon, an. III.)