Search

This paper is divisible into three portions. In the first, the author considers the functions and seat of each of the powers of the living animal; in the second, the nature of each power; and in the third, the manner in which they are associated in the more complicated re­sults which constitute life. Of these powers the simplest is the muscular, which consists merely in a contractile power residing in the muscular fibre itself: and various experiments are referred to in proof that it depends ex­clusively on the state of this fibre, and in no degree on that of the nervous system, which some physiologists have regarded as the real seat of this power: for, instead of being recruited, it is exhausted by the action of the nervous system upon it, as it is by other stimulants.

After a short account of the labours of preceding naturalists in that department of zoology which comprises the various kinds of polypes, and of the different characters on which they have founded the classification of these animals, the author proceeds to the statement of his own observations on several species which had not been previously investigated with sufficient minuteness and care. Two of the species described he believes to be entirely new, and he has accordingly given them the names of Bowerbankia densa , and Lagenella repens . The other species which are the subject of the author’s investigation, are Vesicularia spinosa , Valkeria cuscuta , Alcyonidium diaphanum , Membranipora pilosa , and Notania loriculata .

Hassin et al. [9] consider the Max-Exp-Cover-R problem to study the facility location problem on a graph in the presence of unreliable links when the link failure is according to the Linear Reliability Order (LRO) model. They showed that for unbounded R the problem is polynomial time solvable and for \(R=1\) and planar graphs the problem is NP-Complete. In this paper, we study the Max-Exp-Cover-1 problem under the LRO edge failure model. We obtain a fixed parameter tractable algorithm for Max-Exp-Cover-1 problem for bounded treewidth graphs, parameterized by the treewidth. We extend the Baker’s technique (Baker, J. ACM 1994) to obtain PTAS for Max-Exp-Cover-1 problem under the LRO model on planar graphs. We observe that the coverage function of the Max-Exp-Cover-R problem is submodular and the problem admits a \((1-1/e)\)-approximation for any failure model in which the expected coverage of a set by another set can be computed in polynomial time.

(1840). XLIII. On the detection and estimation of colophony (common rosin) when dissolved in the fixed oils. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 17, No. 110, pp. 289-292.

The effects of the aspect ratio on unsteady solutions through the curved duct flow are studied numerically by a spectral based computational procedure with a temperature gradient between the vertical sidewalls for the Grashof number 100 ⩽ Gr ⩽ 2 000. The outer wall of the duct is heated while the inner wall is cooled and the top and bottom walls are adiabatic. In this paper, unsteady solutions are calculated by the time history analysis of the Nusselt number for the Dean numbers Dn = 100 and Dn = 500 and the aspect ratios 1 ⩽ γ ⩽ 3. Water is taken as a working fluid (Pr = 7.0). It is found that at Dn = 100, there appears a steady-state solution for small or large Gr. For moderate Gr, however, the steady-state solution turns into the periodic solution if γ is increased. For Dn = 500, on the other hand, it is analyzed that the steady-state solution turns into the chaotic solution for small and large Gr for any γ lying in the range. For moderate Gr at Dn = 500, however, the steady-state flow turns into the chaotic flow through the periodic oscillating flow if the aspect ratio is increased.

One of the formalisms to qualitatively describe polylines in the plane are double-cross matrices. In a double-cross matrix the relative position of any two line segments in a polyline is described with respect to a double cross based on their start points. Two polylines are called DC-similar if their double-cross matrices are identical. Although double-cross matrices have been widely applied, a geometric interpretation of the similarity they express is still lacking. In this paper, we provide a first step in the geometric interpretation of this qualitative definition of similarity. In particular, we give an effective characterization of what DC-similarity means for polylines that are drawn on a grid. We also provide algorithms that, given a DC-matrix, check whether it is realizable by a polyline on a grid and that construct, if possible, in quadratic time example polylines that satisfy this matrix. We also describe algorithms to reconstruct polylines, satisfying a given double-cross matrix, in the two-dimensional plane, that is, not necessarily on a grid.

n/a

The principal object of this paper is, to remove the obscurity in which that part of the theory of the curvature of surfaces which relates to umbilical points has been left by Monge and Dupin, to whom, however, subsequently to the labours of Euler, we are chiefly indebted for a comprehensive and systematic theory of the curvature of surfaces. In it the author shows, that the lines of curvature at an umbilic are not, as at other points on a surface, two in number, or, as had been stated by Dupin, limited; but that they proceed in every possible direction from the umbilic. The obscurity complained of is attributed to the inaccurate conceptions entertained by Monge and Dupin, of the import of the symbol 0/0 in the analytical discussion of this question, the equation which determines the directions of the lines of curvature taking the form 0( dy / dx ) 2 + 0( dy / dx ) + 0 = 0 at an umbilic. After stating that Dupin has been guided by the determination of the differential calculus, the author remarks, that in no case is the differential calculus competent to decide whether 0/0, the form which a general analytical result takes in certain particular hypotheses, as to the arbitrary quantities entering that result, has or has not innumerable values. He then states the principle, that those values of the arbitrary quantities (and none else) which render the equations of condition indeterminate must also render the final result, to which they lead, equally indeterminate; and that, therefore, when such result assumes the form 0/0, its true character is to be tested by the equations that have led to it, after these have been modified by the hypothesis from which that form has arisen.

The method of preparing the paper here referred to consists in washing it over with nitrate of silver, then with bromide of potassium, and afterwards again with nitrate of silver; drying it at the fire after each operation. This paper is very sensitive to the light of the clouds, and even to the feeblest daylight. The author supplies an omission in his former memoir on photogenic drawing, by mentioning a method he had invented and practised nearly five years ago, of imitating etchings on copper plate, by smearing over a sheet of glass with a solution of resin in turpentine, and blackening it by the smoke of a candle. On this blackened surface a design is made with the point of a needle, the lines of which will of course be transparent, and will be represented by dark lines on the prepared paper to which it is applied, when exposed to sunshine. The same principle may be applied to make numerous copies of any writing.

The author states, that it has been his object, in this series of investigations, to perfect the methods of electrical measurement, whether relating to the quantity of electricity, intensity, inductive power, or any other element requiring an exact numerical value, and by operating with large statical forces both attractive and repulsive, to avoid many sources of error inseparable from the employment of extremely small quantities of electricity, such as those affecting the delicate balance used by Coulomb. He then describes some improvements in his hydrostatic electrometer, an instrument already mentioned in his first paper, which, although not available for the measurement of such minute forces as those to which the balance of torsion is applicable, is still peculiarly delicate and well adapted to researches in statical electricity. Its indications depending on the force between two opposed planes operating on each other under given conditions, are reducible to simple laws, and are hence invariable and certain; the attractive force between the discs is not subject to any oblique action, is referable to any given distance, and may be estimated in terms of a known standard of weight. The author next proceeds to the further consideration of the subject of his former papers, viz. the elementary laws of electrical action. He proves, by the following experiments, that induction invariably precedes, or at least accompanies attraction and repulsion. A circular disc of gilded wood, about six inches in diameter, is suspended by an insulating thread of varnished silk from a delicate balance; a delicate electroscope is attached to this disc, and the whole is counterpoised by a weight. A similar disc insulated on a glass rod, and having also an electroscope attached to it, is placed at any convenient distance immediately under the former. One of the lower discs being charged with either electricity and the other remaining insulated and neutral, the electroscope of the neutral disc begins to rise, whilst that of the charged disc, already in a state of divergence, tends to collapse: when these respective effects ensue, the suspended disc descends the charged disc. Two inductive actions are indicated in this experiment, the one the author considers to be a direct induction, the other a reflected induction.

The author gives citations from several chronicles of the middle ages descriptive of the remarkable appearance of shooting stars which occurred on the 4th of April, 1095, on the testimony of independent witnesses both in France and England. One of them describes them as “ falling like a shower of rain from heaven upon the earth:" and in another case, a bystander, having noted the spot where the aerolite fell, “cast water upon it, which was raised in steam, with a great noise of boiling.” The Chronicle of Rheims describes the appearance as if all the stars in heaven were driven, like dust, before the wind. A distinct account of the shooting stars of July 26th, 1293, is given by Matthew Paris.

The author describes various experiments made with a view to determine the electro-chemical equivalents of sodium and potassium. Three experiments gave, respectively, 22.3, 22.9, and 25, as the equivalent of the former; and two other experiments gave, respectively, 45 and 41.7, as the equivalent of the latter of these substances. He then inquires what would be the result of the electro-lyzation of the aqueous solutions of soda and potash, on the hypothesis of these bodies being composed of two equivalents, or atoms, of metal, and one of oxygen. To determine this question he employs a solution of dichloride of copper in muriatic acid, as being a substance composed of two atoms of metal and one of an electro-negative element. Its electrolysis gave as the equivalent of copper, 52.8, 59.4, and 61.6, numbers approximating closely to 63.2, or double the atomic weight of copper. After a long train of investigation, he concludes that there is no reason deducible from the theory of isomorphism for doubting the correctness of the received atomic weights of silver, sodium, &c., but that the difficulty, or anomaly, if it may be so called, should be considered as attaching itself to the di-compounds of copper; and that Faraday’s propositions on this subject remain unimpeached.

The object of this paper is to develope a new and remarkably simple method of approximating to the real roots of numerical equations, which possesses several important advantages. After describing the nature of the transformations which are subsequently employed, the author proceeds to develope the process he uses for obtaining one of the roots of a numerical equation. Passing over the difficult question of determining the limits of the roots, he supposes the first significant figure (R) of a root to have been ascertained, and transforms the proposed equation into one whose roots are the roots of the original, divided by this figure (or x /R): one root of this equation lying between 1 and 2, the first significant figure ( r ) of the decimal part is obtained, and the equation transformed into another whose roots are those of the former, divided by 1+ this decimal (or 1 + r ). This last equation is again similarly transformed; these transformations being readily effected by the methods first given. Proceeding thus, the root of the original equation is obtained in the form of a continued product. After applying this method to finding a root of an equation of the 4th, and likewise one of the 5th degree, the author applies it to a class of equations to which he considers it peculiarly adapted, namely, those in which several terms are wanting. One of these is of the 16th degree, having only six terms; and another is of the 622nd degree, having only four terms.

Among various formulæ for the rectification of the circle discovered by the author, he has found the one given in this paper to be that best fitted for computation: and he has been enabled by means of it, with comparatively little labour, to extend the number, expressing the ratio of the diameter to the circumference, to 208 places of decimals, a degree of accuracy hitherto unattainable, without a great amount of labour, by means of any of the series which have yet been employed. The celebrated series of Mr. John Machin, for the rectification of the circle, is derived from the formula π/4 = 4 tan -1 1/5 -tan -1 1/239.

In this communication, the author first demonstrates the well-known property of the circle, that if from a point in the diameter produced there be drawn a tangent to the circle, and from the point of contact there be drawn a line perpendicular to the diameter; and if from any point in the circumference there be drawn two lines, one to the point without the circle, and another to the foot of this perpendicular, the former of these lines will be to the latter, as the distance of the point without the circle from the centre, is to the radius of the circle.

n/a

1. In a paper, which I communicated to the Royal Society in 1839, and which was honoured by insertion in the Transactions of the following year, I described a series of experiments made with some powerful voltaic batteries, for the purpose of determining the possibility of obtaining a spark before the completion of the voltaic circuit. I was therein enabled to establish a few facts respecting polar tension, or rather respecting the absence of any notable degree of it in the batteries I described; for instance, I proved that, with 320 series of Professor Daniell’s constant battery, polar tension was not evinced adequate to the striking distance of 1/5000 th of an inch; nor was I more successful in obtaining it with a water battery of 1024 series, constructed by the same gentleman. I also stated that, according to the present theoretical views of the action of the voltaic battery, with the apparatus I then used, it ought to have taken place; and that, if by a still more powerful apparatus it could not be obtained, the theory must, in some way or other, be incorrect. 2. The preceding negative facts are not without their value in a scientific point of view; they show us, at least, a certain limit within which the anticipated effects could not be obtained. At the same time I could not fail to admit that they were anything but conclusive, as to the actual question of the possibility of obtaining the spark before the circuit was completed. That I am justified in calling the spark, under such circumstances, an anticipated effect, may be fairly assumed, because every electrician is aware that the terminals of a voltaic series invariably evince a certain amount of tension and as spark is but a consequence of tension exalted to a maximum, it is only fair to anticipate that, by increasing the tension, it would be obtained.

This paper is intended as a necessary supplement to the paper bearing the same title already printed in the Society's Transactions, though but an abridgment of a larger one which the author had prepared on the subject. Particular circumstances induced him to alter the plan he had originally contemplated, and instead of a complete development in detail of his researches and his views, he has only on the present occasion given so much of his results as were necessary to bring the system of polar spherical co-ordinates to a state analogous to that in which plane polar curves has long been stationary, one point of the analogy excepted, viz. where the author has extended the method of treating tangents and normals, and the consequent investigations dependent on these, by giving the polar equations of those lines, instead of merely examining the relation between the radius-vector, and perpendicular upon the tangent. In a note the equations of the tangent and normal, to plain curves, is given from first principles; and the analogy between plane and spherical curves is shewn to be remarkably close.

The object of this paper is to determine in what consists the difference between general Symbolical Algebra and the sciences subordinate to it, particularly Arithmetical Algebra. The view which the author takes is, that Symbolical Algebra takes cognizance only of the laws by which the symbols are combined, without considering the nature of the operations represented. The greater part of the paper is occupied in applying this definition, by shewing what are the laws to which are subject the various symbols of operations we are in the habit of using; and one or two examples are given of the advantages derivable from this way of considering the subject—particularly with respect to the connection between the arithmetical and geometrical meanings of + and −. The chief application of the theory may be said to be the elucidation of the causes of analogies between operations by no means similar in their nature.

The author first adverted to the discovery, nearly about the same time, of paraffine by Reichenbach and of petvoline by Dr Christison. The former occurred among the products of destructive distillation; the latter was found in the Rangoon petroleum, and they were soon found to be identical. Reichenbach's researches on naphtha were then quoted, by which it appears that that indefatigable observer could not discover, in the kind of naphtha which he examined, any trace either of paraffine, or of any other product of destructive distillation. On the contrary, he found that naphtha to possess the characters of oil of turpentine, a product of vegetable life j and he succeeded in obtaining a precisely similar oil from brown coal by distillation at 212°.

Various accounts have been given of the alcoholic strength of wines by Mr Brande, Julia-Fontenelle, and others. The author has been engaged for some time in experiments for determining the proportion of alcohol contained in various wines of commerce, and also the circumstances which occasion a variety in this respect. The present paper is an interim notice of the results.The method of analysis consisted in the mode by distillation, which was applied with such contrivances for accuracy that nearly the whole spirit and water were distilled over without a trace of empyreuma, and without the loss of more than between 2 and 6 grains in 2000. From the quantity and density of the spirit, the weight of absolute alcohol of the density 793.9, as well as the volume of proof spirit of the density 920, was calculated from the tables of Richter founded on those of Gilpin.

The object of this paper is to investigate, for the ellipse and hyperbola, a series of propositions analogous to those which, in the circle, constitute the calculus of sines.Any diameter of an ellipse, or a transverse diameter of a hyperbola, is expressed by a, its conjugate diameter by b, and the ratio of these diameters, viz. by c. The nature of the curves is exbpressed by this formula, In the ellipse, c has the sign +, but in the hyperbola the sign −.

The functional equation to be resolved is this—In this equation xo° and x′ denote any two values of the indefinite variable x, and c is a constant quantity. The object of inquiry is the forms of the function.It is proved that there are two forms of the function, which alike satisfy the proposed equation: viz.In the first of these functions x increases, while f(x) decreases; and in the second, x and f(x) increase together. The first of these functions is applied in the paper to the theorem of the Parallelogram of Forces, whilst the theory of Curves of Equilibration is deduced from the second function: these last, in their most general form, have two parameters. The curve of equilibration is formed by a perfectly flexible chain suspended in a vertical plane between two fixed points.

In this paper, a procedure of D-K iteration with both frequency and constant scalings for scaled H ∞ synthesis to systems with both L 2 norm bounded linear and nonlinear uncertainties is proposed. In each step, the procedure consists of two LMI optimizations and a standard H ∞ , control design.

Shor's factoring algorithm (SFA) finds the prime factors of a number, $N=p_1 p_2$, exponentially faster than the best known classical algorithm. Responsible for the speed-up is a subroutine called the quantum order finding algorithm (QOFA) which calculates the order -- the smallest integer, $r$, satisfying $a^r \mod N =1$, where $a$ is a randomly chosen integer coprime to $N$ (meaning their greatest common divisor is one, $\gcd(a, N) =1$). Given $r$, and with probability not less than $1/2$, the factors are given by $p_1 = \gcd (a^{\frac{r}{2}} - 1, N)$ and $p_2 = \gcd (a^{\frac{r}{2}} + 1, N)$. For odd $r$ it is assumed the factors cannot be found (since $a^{\frac{r}{2}}$ is not generally integer) and the QOFA is relaunched with a different value of $a$. But a recent paper [E. Martin-Lopez: Nat Photon {\bf 6}, 773 (2012)] noted that the factors can sometimes be found from odd orders if the coprime is square. This raises the question of improving SFA's success probability by considering odd orders. We show that an improvement is possible, though it is small. We present two techniques for retrieving the order from apparently useless runs of the QOFA: not discarding odd orders; and looking out for new order finding relations in the case of failure. In terms of efficiency, using our techniques is equivalent to avoiding square coprimes and disregarding odd orders, which is simpler in practice. Even still, our techniques may be useful in the near future, while demonstrations are restricted to factoring small numbers. The most convincing demonstrations of the QOFA are those that return a non-power-of-two order, making odd orders that lead to the factors attractive to experimentalists.

In this paper, we focus on the identification of discrete-time piecewise affine (PWA) systems from noisy data. This problem consists of the estimation of both the local affine subsystems and the partition of the regression space. A two-stage robust identification approach is proposed to estimate the local affine subsystems in the presence of noise. This approach includes an optimization-based initial estimation process and a least-squares-based refinement procedure. In addition, to estimate the partition of the regression space for continuous dynamic PWA systems, an intersection approach is proposed as an alternative to the widely used pattern recognition approaches. Simulation studies demonstrate the effectiveness of the two-stage identification approach and the intersection approach in noisy case.

Bergstra, Bethke and Ponse proposed an axiomatization for Basic Process Algebra extended with (binary) iteration. In this paper, we prove that this axiomatization is complete with respect to strong bisimulation equivalence. To obtain this result, we will set up a term rewriting system, based on the axioms, and prove that this term rewriting system is terminating, and that bisimilar normal forms are syntactically equal modulo commutativity and associativity of the +.

(1847). Note on the means of testing the comparative value of astringent substances for the purposes of tanning. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 31, No. 206, pp. 150-151.

This paper is devoted to the study of the oscillation of solutions of delay differential equations of the neutral and mixed types. Some general results are proved for certain general Volterra type neutral differential equations and many particular cases are discussed.

n/a

The investigations presented in this paper consist of two parts; the first offers a solution, in a certain qualified sense, of the general linear equation in finite differences; and the second will be found to give an almost complete analysis of the resolution in series of the general linear differential equation with rational factors. The second part is deduced directly from the results of the first, although the subjects of which they respectively treat appear to be wholly independent of each other. With the exception of a few cases capable of solution by partial and artificial methods, there does not at present exist any mode of solving linear equations in finite differences of an order higher than the first; and with reference to such equations of the first order, we are obliged to be content with those insufficient forms of functions which are intelligible only when the independent variable is an integer, and which may be obtained directly from the equation itself by merely giving to the independent variable its successive integer values. It is in this insufficient and qualified sense that the solutions here given are to be taken ; and the first part of the following investigations may be considered as an extension of this form of solution from the general equation of the first order to the general equation of the with order.

(1.) Before entering on the investigation which forms the object of this communication, it will be necessary to recall to recollection some general properties of the differences of the powers of the natural numbers, or of the numbers comprised in the general expression ∆ m 0 n , which I have elsewhere demonstrated, as well as to establish certain preliminary theorems by the aid of those properties, which will be useful in the progress of the inquiry. I shall employ throughout the separation of the symbols of operation from those of quantity, as respects ∆ and 0, in the manner followed in my paper “ On the Development of Exponential Functions,” published in the Philosophical Transactions, vol. cvi. p. 25 (1816), and further extended in its appli­cation in my “ Collection of Examples of the Applications of the Calculus of Finite Differences,” appended to the translation of Lacroix’s Differential and Integral Cal­culus in 1820, to which paper and collection the reader is referred for the demon­stration of the fundamental properties in question. (2.) Denoting by F( x ) any series of powers of x , such as F ( x )=A x a +B x b +C x c +&c., and by f ( x ) any other as f ( x )=P x p +Q x q +R x r +&c.

Until recently, general solutions of several classes of equations, such as that which occurs in the theory of the figure of the earth, could not be arrived at. An ingenious transformation lately rendered it a matter of comparative ease to arrive at a solution of these equations in those forms in which they are presented in the solution of physical problems; but still much remains to be done. The object of the present paper is to supply some portion of the deficiency in this respect, by the introduction of a new transformation, and the adoption of the function Γ. The solutions thus obtained are perfectly general, and are arrived at with the greatest facility.

The author commenced his paper by referring to the analysis of codeine made by different chemists. On these analyses four different formulæ had been founded; but two only, those of Regnault and of Gerhardt, required special mention, the others being now known certainly not to represent the constitution of the base. Regnault had deduced from his analysis the formula C35H20NO5, while Gerhardt gives C36H21NO6as the expression of his results.

During the last session of the Institute of Actuaries, I had the honour to read a paper on the Value of the Principle of Selection as exercised by the Offices in granting Assurances upon Lives, and to submit some calculations which I had made with reference to that subject, in which calculations the lives admitted at particular ages were kept apart from others during their subsequent existence. I have now to offer to the Institute an extended set of Tables of a similar character; and, after stating the course which I have followed in constructing them, I shall make a few observations on the value of selection as exercised by the policy-holder against the Company.

An assertion made by Sir Isaac Newton in a letter to Conti, published in Raphson’s History of Fluxions, that the materials of the Commercium Epistolicum were “ collected and published by a numerous Committee of gentlemen of different nations , appointed by the Royal Society for that purpose,” appeared to be at variance with the list of the Committee as it was appointed on the 6th of March, 1711- 12, and which only contains the names of Arbuthnot, Hill, Halley, Jones, Machin and Burnet, who were all English. But on further search of the records of the Society with the aid of Mr. Weld, the Assistant Secretary, the author ascertained that other members were subsequently added to the Committee, among whom were Bonet, the Prussian minister, and De Moivre, both of whom were foreigners ; thus showing that the imputations which might have been cast on Newton’s veracity are groundless.

The author of this paper, after noticing Wilson’s Theorem, (pub­lished by Waring about the year 1770, without any proof), which theorem is that, if A be a prime number, 1. 2. 3. . . . (A —1)+1 is divisible by A ; refers to Lagrange’s and Euler’s demonstrations, and mentions Gauss’s extension of the theorem, to any number, not prime; provided that instead of 1, 2, 3, &c. (A —1), those numbers only be taken which are prime to A, and 1 be either added or sub­tracted. This theorem was published by Gauss without a proof in 1801, with a rule as to the cases in which 1 is to be added or sub­tracted, the correctness of which is questioned by the author, who proceeds to propound the following theorem, which he had previ­ously, for distinctness, divided into three. If any number, prime or not, be taken, and the numbers prime to it, and less than one half of it be ascertained, and those be rejected whose squares ±1 are equal to the prime number, or some multiple of it (which may be more than one), then the product of the re­maining primes (if any), ±1 shall be divisible by the prime number.

Having been recently engaged in a series of experiments on the fermentation of sugar and molasses, which rendered it necessary to refer to a table of the value in absolute alcohol of spirits of different specific gravities, the author found himself compelled to construct for this purpose a new table, which he lays before the Royal Society in the present paper. The table was formed by weighing out absolute alcohol and distilled water in fixed proportions, mixing them, and after allowing time for condensation, determining with suitable precautions the specific gravity of each mixture at the temperature of 60° Fahrenheit. Each alternate number in the table was so obtained; the rest being interpolated. The alcohol employed was prepared by digesting the strongest rectified spirit, first with dry carbonate of potash and afterwards with powdered quicklime and distilling. It had the specific gravity ·7938 at 60°, and boiled at 177° Fahr.

1. By the aid of two simple theorems expressing the laws under which the operations of differentiation combine with operations denoted by factors, functions of the independent variable, the author arrives at a principle extensively applicable to the solution of equations, which may be stated as follows:—“if any linear equation φ ( x , D). u = X have for its solution u = ψ ( x , D). X, this solution being so written that the operations included under the function ψ are not performed or suppressed, then φ (D, - x ). u = X has for its solution u = ψ (D, - x ). X.” The solution thus obtained may not be, and often is not, interpretable, at least in finite terms; but if by any transformation a meaning can be attached to this form, it will be found to represent a true result. An important solution immediately deducible from this principle is given by Mr. Boole in the Philosophical Magazine for February 1847, and is extensively employed in the present paper. It is immediately obtained by making the conversion above proposed in the general equation of the first order and its solution.

This paper professes to investigate certain properties of the series of whole numbers whose ultimate differences are constant, and incidentally to treat of Fermat’s theorem of the polygonal numbers, and some other properties of numbers. Its object is to show that the same (or an analogous) property which Fermat discovered in the polygonal numbers belongs to other series of the same order, also to all series of the first order, and probably to all series of all orders. It also proposes to prove the first case of Fermat’s theorem (that is of the triangular numbers) from the second case of the squares (which had not before been done), and to dispense with the elaborate proof of Legendre (Théorie des Nombres), finally, to prove all the cases by a method different from that either of Lagrange, Euler, or Legendre.

The object of this communication is to diminish the labour attendant upon existing methods for the analysis of numerical equations. As Budan pointed out intervals, within the bounds of the extreme limits of the roots of an equation, in which all search for roots would be fruitless, so here the author seeks for what he terms “ rejective intervals ” among those which Budan had retained. This he proposes effecting by transforming the first member of every equation X = 0 into X = {F + √¯F 2 X} x {F — √¯F 2 - X } .....................(1.) which the author calls decomposing it into conjugate factors ; in which factors, F is entirely arbitrary. On account of this unrestricted character of F, innumerable changes may be effected on a pair of conjugate factors, without disturbing their product: but it is from the following expression of these factors that the general results in this paper are chiefly deduced: namely, X = { F + ∫ + √¯F 2 + 2F ∫ + ∫ 2 - X } } × { F + ∫ - √¯F 2 + 2F ∫ + ∫ 2 - X } }...............(2.)

The object of this paper professes to be to ascertain whether the principle of Fermat’s theorem of the polygonal numbers may not be extended to all orders of series whose ultimate differences are con­stant. The polygonal numbers are all of the quadratic form, and they have (according to Fermat’s theorem) this property, that every number is the sum of not exceeding, 3 terms of the triangular num­bers, 4 of the square numbers, 5 of the pentagonal numbers, &c. It is stated in this paper that the series of the odd squares 1,9,25,49, &c. has a similar property, and that every number is the sum of not exceeding 10 odd squares. It is also stated, that a series con­sisting of the 1st and every succeeding 3rd term of the triangular series, viz. 1,10,28,35, &c., has a similar property; and that every number is the sum of not exceeding 11 terms of this last series, and that this may be easily proved [it was proved in a former paper by the same author]. The term “Notation-limit” is applied to the num­ber which denotes the largest number of terms of a series necessary to express any number; and the writer states that 5,7,9,13,21 are respectively the notation-limits of the tetrahedral numbers, the octa­hedral, the cubical, the eicosahedral and the dodecahedral numbers; that 19 is the notation-limit of the series of the 4th powers; that 11 is the notation-limit of the series of the triangular numbers squared, viz. 1,9,36,100, &c., and 31 the notation-limit of the series 1,28,153, &c. (the sum of the odd cubes), whose general expression is 2 n 4 — n 2 .

The methods employed in this paper to effect the solution or reduction of linear differential equations consist of certain peculiar transformations, and each particular class of equations is transformed by a distinct process peculiarly its own. The reduction is effected by means of certain general theorems in the calculus of operations. The terms which form the first member of the first class of equations are functions of the symbols ɯ and τ, the latter being a function of x , and the former a function of x and D, x being the independent variable. This member of the equations contains two arbitrary functions of vs, and may therefore be of any order whatever. It likewise contains two simple factors, such for example as ɯ+ nk and which factors are taken away by the transformation employed, and consequently the equation is reduced an order lower; it is therefore integrated when of the second order. There is a series of equations of this class, each essentially distinct from the rest, yet all reducible by a similar process.

The object of this paper is to exemplify and extend the mode of analysis explained by the author in two former communications to the Royal Society “ On the Development of Exponential Functions,” and “ On Circulating Functions,” to a case in the theory of numbers in which they afford remarkable facilities, viz. that of the partitions of which a given number is susceptible. The separation of the symbols of operation from those of quantity, in the mode explained in the former of those communications, allows of the expression of the sums of certain series entering into this theory, under a form susceptible of resolution (by a theorem here given) into two portions, one of which, a rational function of the independent variable or number to be subdivided, expresses approximately, as a rational fraction, the number of partitions; the other, a periodic or circulating function, expresses the fraction, less than unity, by which the other portion differs from an exact integer number, and which, applied with its proper sign to that former portion, reduces it to an integer. In the mode of procedure here followed all subdivision into cases according to the numerical constitution of the number to be subdivided is avoided, and a uniform treatment is carried throughout.

65. In the course of some researches upon inverse problems regarding distributions of magnetism, and upon the comparison of electro-magnets and common magnets, I have found it extremely convenient to make use of definite terms to express certain distributions of magnetism and forms of magnetized matter possessing remarkable properties. The use of such terms will be of still greater consequence in describing the results of these researches, and therefore, before proceeding to do so, I shall give definitions of the terms which I have adopted, and explain briefly the principal properties of the magnetic distributions to which they are applied. The remainder of this chapter will be devoted to three new methods of analysing the expressions for the resultant force of a magnet at any point, suggested by the consideration of these special forms of magnetic distribution. A Mathematical Theory of Electro-Magnets, and Inverse Problems regarding magnetic distributions, are the subjects of papers which I hope to be able to lay before the Royal Society on a subsequent occasion. 66. Definitions and explanations regarding Magnetic Solenoids (1.) A magnetic solenoid is an infinitely thin bar of any form, longitudinally magnetized with an intensity varying inversely as the area of the normal section in different parts.

If we consider the very different forms which the solutions of Differential Equations differing very little from each other frequently take, and the very different processes often required in each particular case to obtain the solution, we shall be led to conclude that the discovery of any universal or general method of solving them must be a hopeless case. We cannot therefore regard particular methods, especially when applicable to a large number of cases, as useless speculations. The present paper contains the solution of several classes of these equations effected by means of general theorems in the Calculus of Operations adapted to each particular class. For expla­nation of the symbols employed, let it be observed that D is put for and that d/dx , and that φ, λ, ψ, and X denote any functions of x , the independent variable, and are the same as φ( x ), λ( x ), &c.; and in like manner φ(D), λ(D), &c. will be used to denote the same functions of D.