William Fishburn Donkin's publications

We list below those publications of William Fishburn Donkin that we have been able to find. For most we give a brief extract to indicate the nature of the work.

W F Donkin, An Essay on the Theory of the Combination of Observations (The Ashmolean Society, Oxford, 1844).

Presented to the Ashmolean Society, 26 February 1844.

It is well known that the method of Least Squares does not, strictly speaking, give the most probable result to be drawn from a system of observations, unless either the number of observations be indefinitely great, or the function expressing the relative facility of errors have a particular form; whilst it is certain that, in practice, neither of these conditions ever subsists otherwise than approximately.

Yet it seems to be admitted that the method in question does strictly, in some sense or other, give the best result which can be obtained. No one, however. attempted a demonstration of this principle before Professor Gauss. In his treatise entitled "Theoria Cornbinationis Observationum erroribus minimis obnoxiae," (Gottingen, 1828), he has considered the following question: Admitting that the number of observations is finite, and that we do not know the form of the function by which their character is defined, what conclusion ought we to adopt under these circumstances? or, to employ his own term, what is the most plausible result? And he has demonstrated that the method of Least Squares must be used, provided we assume that result to be most plausible which would give the least average value to the square of the error in an infinite number of trials. (Theor. Comb. art. 6. and 21.) But this assumption, however strongly it may commend itself to the instinct of the mathematician, confessedly involves something arbitrary, and so there is still something wanting to the logical consistence of the theory which rests upon it.

In the following essay the subject is considered under a different aspect, but the results are identical with those obtained in the treatise just referred to, at least so far as they are comparable with them. The reasoning employed professes to be demonstrative in the same sense in which the ordinary a priori proofs of the fundamental theorems of Statics profess to be so; and as it is important that it should be clearly understood in what sense this is, I have added some remarks at the end upon this point. But whatever may be thought of the logical character of the processes, I believe it will be allowed that they contain nothing arbitrary; that no principle is assumed which is not obviously and unquestionably the most plausible, or rather the only plausible one which could be assumed at all. And if so, the complete coincidence of the conclusions with those deduced from a perfectly distinct and independent set of principles may be looked upon as an interesting fact, notwithstanding the failure of any particular attempt to explain it.
W F Donkin, On the Rhythm of Ancient Greek Music, The Classical museum: a journal of philology, and of ancient history and literature, London 2 (1845), 110-115.

In modern music there is melody, harmony, time, and rhythm. In ancient Greek music there was melody, certainly; harmony, probably not; time there must have been, by the same necessity that there is space in painting; and if rhythm be proportion of times, then there must have been rhythm too, as there must be some proportion between the lines in a picture. But was there rhythm, or time, in the modern technical sense? In modern music rhythm and time are not the same thing: at least there are two distinct things which may conveniently be called by those names. A melody is in time if the duration of each sound (or, silence) bear a determinate and very simple proportion to the duration of every other. There is no modern music which is not in time in this sense. For whatever license the performer may take to himself, still it is a licence, and the hearer feels that it is, and continues, in spite of it, to trace the proportions of the times, altering, as it were, the unit of the scale by which he measures them. It is like measuring lines with an elastic rule, and so making them seem to have the proportion we desire. The idea of strict time is never relinquished for a moment, either by the musician or the bearer. But rhythm is something more than time. Music may be in time without being rhythmical; it cannot be rhythmical without being in time. It is in time, when the duration of every sound is, actually or in idea, expressible in simple numbers by means of the same unit. It is rhythmical, when the sounds are distributed into regular groups or periods, marked by the recurrence of accents ...
W F Donkin, On the geometrical laws of the motion of a rigid system about a fixed point, London Edinburgh and Dublin Philosophical Magazine and Journal of Science (3) 36 (245) (1850), 427-433.

There is a very simple theorem which seems to be capable of useful applications in the theory of the motion of a rigid system. I do not remember ever to have seen it explicitly stated, though Mr Boole appears to refer to it in a paper published some time ago in this Journal, which will be mentioned below. I propose here to give a geometrical demonstration of it, after establishing the necessary definitions and conventions.

In considering the motion of a rigid system about a fixed point, let us conceive a sphere to be described about that point as a centre, and fixed in space; and another sphere, with the same centre and radius, to be invariably connected with and carried by the moveable rigid system; so that the motion of this latter sphere may be substituted for that of the rigid system, so far as its geometrical displacement only is concerned. For convenience, let the radius of each sphere be taken equal to the linear unit.

When a rotation takes place about any axis passing through the centre, let the points of intersection of that axis with either sphere he called the poles of the rotation; and let that be called the positive or north pole, with respect to which the rotation has the same direction as the earth's diurnal motion with respect to its north pole.
W F Donkin, On certain theorems in the calculus of operations, The Cambridge and Dublin Mathematical Journal 5 (1850), 10-18.

It seems from these investigations, and from the researches of Mr Boole, Mr Bronwin, and others, in the same department, that there is much more analogy than might have been expected a priori between the laws of commutative and non-commutative symbols. It appears probable also that in the further progress of inquiry some extension of notation may become necessary, or at least convenient.
W F Donkin, On the geometrical interpretation of quaternions, London Edinburgh and Dublin Philosophical Magazine and Journal of Science (3) 36 (246) (1850),489-503.

In a recent paper in this Journal, I gave two examples of the application of quaternions to geometrical problems, in which I made use of a system of interpretation different from that usually employed by Sir W Hamilton. This system it is my present object to explain. I am not at present prepared to offer an opinion as to its practical advantages or disadvantages, It is here proposed as possessing a certain theoretic interest, arising from the fact that it represents the method of quaternions as a natural (or rather the natural) extension to tridimensional space of the usual geometrical interpretations of symbolical algebra. The general principles employed in the investigation have no pretensions to novelty. The view here taken of symbolical algebra, and of its interpretation in plane geometry, is substantially the some as that advocated by the late Mr Gregory (in the fourteenth volume of the Edinburgh Philosophical Transactions, and in several papers in the Cambridge Mathematical Journal); and the extension to geometry of three dimensions rests upon principles which will be found, I believe, to agree essentially, us far as they go, with those laid down by Sir W Hamilton in his papers on symbolical geometry in the last-named periodical. In what follows, then, no further reference will in general be made to previous writers. Moreover, to avoid the use of frequent qualifying phrases, propositions will be stated dogmatically which only claim to be accepted as belonging to a consistent system, not as belonging to the only consistent or useful system.
W F Donkin, On certain questions relating to the theory of Probabilities, London Edinburgh and Dublin Philosophical Magazine and Journal of Science (4) 1 (5) (1851), 353-368.

Within the last year three papers have been published on subjects connected with the theory of probabilities, all discussing or suggesting questions of more or less importance and difficulty; namely, an article in the Edinburgh Review for July 1850; remarks on certain part of this article by Mr R L Ellis, in the last November Number of this Journal; and a paper by Prof J D Forbes in the December Number of the same. In what follows, I am not going to attempt anything like a detailed examination of any of these papers, but merely to offer some considerations which may, I hope, contribute something, however little, towards a settlement of the questions treated in them, so far as they involve matters of doubt or controversy. How far there may be any originality, either of substance or form, in anything that I shall say, I hope I may be allowed to leave the reader to settle for himself, if he care about it; though of course I shall not intentionally make unacknowledged use of the labours of others, unless in so far as their results may be considered to have become common property.
W F Donkin, On certain questions relating to the theory of Probabilities - Part II, London Edinburgh and Dublin Philosophical Magazine and Journal of Science (4) 1 (5) (1851), 458-466.

In a former paper I gave a brief sketch of the general principles of the theory of probabilities considered under the aspect which seemed most advantageous with reference to its practical applications. I also discussed some problems with a view to exhibit the accordance between the results of the theory and the conclusions of common sense. In continuing the subject, I will begin by noticing the important distinction, to which Professor Forbes has drawn attention, between what may be called improbability and incredibility; that is, between the difficulty of believing beforehand that an assigned event will happen, and the difficulty of believing afterwards that a recorded event has happened. The distinction is perfectly well known, and recognised in the mathematical theory; but there are some points connected with it which it is worth while to examine a little in detail. Professor Forbes's experiment with the grains of rice will furnish an illustration.
W F Donkin, On certain questions relating to the theory of Probabilities - Part III, London Edinburgh and Dublin Philosophical Magazine and Journal of Science (4) 2 (8) (1851), 55-60.

I propose in this third and last communication, to offer a few remarks on the method of least squares, chiefly with reference to Mr Ellis's paper on that subject in the Philosophical Magazine for November 1850.

If we are asked what is the method of obtaining the most probable result from a system of observations not numerous enough to justify, as an approximation, the supposition that they are infinite in number, it is plain that no answer can be given till we are told whether it is to be assumed that the law or laws of facility of errors in the individual observations are known, or unknown; or, to speak more accurately, until we are told what is to be assumed as the state of information of the observer concerning the laws in question. For the probability of every hypothesis depends upon the state of information presupposed concerning it.

If the law of facility of errors (which we will suppose, for simplicity, the same in all the observations) be assumed as known, the problem involves no difficulty of principle, though for most laws the required integrations would be impracticable.

But if the law be wholly or partially unknown, though it is still easy to indicate the way in which the problem ought, theoretically, to be treated, the processes required are, in all actual cases, entirely beyond the present powers of analysis.
W F Donkin, On the geometrical theory of rotation, London Edinburgh and Dublin Philosophical Magazine and Journal of Science (4) 1 (3) (1851), 187-192.

Although two demonstrations have already appeared in this Journal of the triangle of rotations (to adopt Mr Sylvester's convenient designation), I think it worth while to add the following, which, through not substantially different, exhibits the theorem in the simplest way, and under the most striking aspect. I shall subjoin some further illustrations of its use in connection with quaternions.
W F Donkin, On a class of differential equations, including those which occur in dynamical problems.- Part I, Philosophical Transactions of the Royal Society 144 (1854), 71-113.

The analytical theory of dynamics, as it exists at present, is due mainly to the labours of Lagrange, Poisson, Sir W R Hamilton, and Jacobi; whose researches on this subject present a series of discoveries hardly paralleled, for their elegance and importance, in any other branch of mathematics.

The following investigations in the same department do not pretend to make any important step in advance; though I should not of course have presumed to lay them before the Society, if I had not hoped they might be found to possess some degree of novelty and interest.

Of previous publications with which I am acquainted, those most nearly on the same subject are, Sir W R Hamilton's two memoirs "On a General Method in Dynamics" in the Philosophical Transactions; Jacobi's Memoir in the 17th vol. of Crelle's Journal, "Ueber die Reduction der partiellen Differential-gleichungen," etc.; and M Bertrand's "Memoire sur Integration des equations differentielles de la Mécanique," in Liouville's Journal (1852). The relation in which the present essay stands to the papers just named will be apparent to those who are acquainted with them, and it would be useless to attempt to make it intelligible to others.
...
I shall here conclude this part of the subject, as it would be beyond the scope of this essay to enter into the details of any of the various problems which might be taken in illustration of the theory, such as those which relate to precession and nutation, or to the motion of the moon about its centre of gravity. The investigations of this section have been introduced, because the results, so far as they go, appeared interesting in themselves, and afforded a remarkable example of the application of the general method.

P.S. Since the last sheets of this essay were in type, I have seen for the first time two papers by Professor Brioschi, in Tortolini's Annali for August and October 1853, of which the titles are "Sulla variazione delle costanti arbitrarie nei problemi della Dinamica," and "Intorno ad un teorema de Meccanica." I have not had an opportunity of examining them sufficiently to judge how far any of the preceding investigations may have been anticipated in them.
W F Donkin, On a class of differential equations, including those which occur in dynamical problems.- Part II, Philosophical Transactions of the Royal Society 145 (1855), 299-358.

The following paper forms the continuation and conclusion of one on the same subject presented to the Royal Society last year, and printed in the Philosophical Transactions for 1854. I have however put it, as far as possible, in such a form as to be independently intelligible.

The fourth Section (the first of this Part) contains a recapitulation of some of the most important results of the former Part, in the form of seven theorems, here enunciated without demonstration.

In the fifth Section the method of the variation of elements is treated under that aspect which belongs to it in connexion with the general subject. It is applied, by way of example, to deduce the expressions for the variations of the elliptic elements of a planet's orbit from the results of art. 30 (Part I.), on undisturbed elliptic motion; this example was chosen, partly because the resulting expressions are required in a future section, and partly for the sake of incidentally calling attention to a fallacy which has been, perhaps, often committed, and certainly seldom noticed. The same method, under a slightly different and possibly new point of view, is applied, as a second example, to the problem of the motion of a free simple pendulum, omitting the effect of the earth's rotation. I believe the methods of this paper might be advantageously employed in the treatment of that general form of the problem of a free pendulum which has been considered by Professor Hansen in his Prize Essay. I was unwilling, however, to attempt what might have turned out to be merely an unconscious plagiarism, without having seen the Essay in question, of which I only succeeded in obtaining a copy on the day of writing this preface. As I now perceive that the investigation would be quite independent, I hope to enter upon it at some future time.

The sixth Section contains some general theorems concerning the transformation of systems of differential equations of the form considered in this paper, by the substitution of new variables. The most important case consists in the transformation from fixed to moving axes of coordinates, in dynamical problems. Some of the results are, I think, interesting, and perhaps new.

The seventh and last Section contains an application of the preceding theorems, in connexion with the variation of elements, to the transformation of the differential equations of the planetary theory. This investigation, if interesting at all, will probably be so to the mathematician rather than to the astronomer. I think, however, that if the theories of physical astronomy were more frequently treated rigorously and symmetrically, apart from any approximate integrations; and if, when the latter are introduced, more care were taken to give a clear and exact view of the nature of the reasoning employed, it might be possible to draw the attention and secure the cooperation of a class of mathematicians who now may well be excused, if, after a slight trial, they turn from the subject in disgust, and prefer to expatiate in those beautiful fields of speculation which are offered to them by other branches of modern geometry and analysis.

The contents of the two last Sections are more or less closely connected with the subjects of various memoirs by other writers, especially Professor Hansen and the Rev B Bronwin. I cannot pretend to that degree of acquaintance with them which would enable me to give an exact statement of the amount of novelty to be found in my own researches. I believe it is enough to justify me in offering them to the Society; beyond this I make no claim.
W F Donkin, On a class of differential equations, including those which occur in dynamical problems, Proceedings of the Royal Society of London 7 (1856), 4-7.

This paper is intended to contain a discussion of some properties of a class of simultaneous differential equations of the first order, including as a particular case the form (which again includes the dynamical equations),
$x'_{i} = \Large\frac{dZ}{dy'_{i}}\normalsize y'_{i} = - \Large\frac{dZ}{dx_{i}}$ , ... (I)
where $x_{1} ... x_{n}, y_{1} ... y_{n}$ are two sets of $n$ variables each and accents denote total differentiation with respect to the independent variable $t$ ; $Z$ being any function of $x_{1}$ etc., $y_{1}$ etc., which may also contain $t$ explicitly. The part now laid before the Society is limited to the consideration of the above form.

After deducing from known properties of functional determinants a general theorem to be used afterwards, the author establishes the following propositions ...
W F Donkin, On a class of differential equations, including those which occur in dynamical problems - Part II, Proceedings of the Royal Society of London 7 (1856), 314-316.

This is the second and concluding part of a paper of which the first part was printed in the Philosophical Transactions for 1854. In the fourth section (the first of this part) some of the most important results of the former part are recapitulated.

In the fifth section the theory of the Variation of Elements is considered under that aspect which belongs to it in connexion with the general methods of this paper; and the facility of its application is shown in two instances: (1) the expressions for the variations of the elliptic elements of a disturbed planet's orbit are deduced from the results of Art. 30 (Part I.), on undisturbed elliptic motion; (2) the problem of determining the motion of a free simple pendulum (omitting the effect of the earth's rotation) is treated by considering the orbit of the projection of the bob upon a horizontal plane as a disturbed ellipse. The differential equations which define the variations of the elements of the ellipse are given in a rigorous form, and integrated approximately so as to give the motion of the apsides of the mean ellipse in any case where the pendulum never deviates much from the vertical, and the motion is not very nearly circular. The result agrees with the conclusions of the Astronomer Royal (Proceedings of the Royal Astronomical Society, vol. xi. p. 160).
W F Donkin, On the equation of Laplace's functions, etc., Philosophical Transactions of the Royal Society 147 (1857), 43-57.

This equation [considered in this paper] was first solved in finite terms by Mr. Hargreave, but in a form very inconvenient for practical applications. A solution free from this objection was after wards obtained by Professor Boole, by a method explained in his memoir "On a General Method in Analysis," Philosophical Transactions, 1844. Lastly, in the second volume of the Journal referred to, the same mathematician gave two more solutions, one of which however is reducible, as he states, to Mr Hargreave's form; the other, though much more convenient than this, is still for most purposes probably less useful than that given in the first volume, from which it differs essentially in form, as well as in the method by which it is deduced.

In the following pages I have treated the equation by a very simple method. The result bears a general resemblance to Professor Boole's first solution, and I conceive that the two forms must be capable of being identified by the assumption of a proper relation between the arbitrary functions; but I am not able at present to show this identity.
W F Donkin, On the equation of Laplace's functions, etc. (Abstract), Proceedings of the Royal Society of London 8 (1857), 307-310.
W F Donkin, On the Analytical Theory of the Attraction of Solids Bounded by Surfaces of a Class Including the Ellipsoid. [Abstract], Proceedings of the Royal Society of London 10 (1859-60), 180-183.
W F Donkin, On the Analytical Theory of the Attraction of Solids Bounded by Surfaces of a Class Including the Ellipsoid, Philosophical Transactions of the Royal Society 150 (1860), 1-11.

The following investigation is the result of an attempt to simplify the analytical treatment of the problem of the Attraction of Ellipsoids. The application to this particular case, of certain known propositions relating to closed surfaces in general, showed that the principal theorems could easily be deduced without taking account of any other properties of the ellipsoid than those expressed by two differential equations, of which the truth is evident on inspection.
W F Donkin, On the secular acceleration of the moon's mean motion, Monthly Notices of the Royal Astronomical Society 21 (8) (1861), 221-228.

The controversy respecting the Moon's acceleration is one upon which no mathematician, professionally connected with Astronomy, could be content to accept without examination the conclusions of others. For my own satisfaction, therefore, I undertook the following investigation of the coefficient of $m^{4}$ , the calculation of which involves the whole mathematical question. For this purpose I have used the method of the variation of elements, which bas already been applied by M Delaunay; but as I have not had an opportunity of seeing what he has on the subject, my own result is quite independent. It will be seen that the value which I obtain for the term in question agrees with that given by Professor Adams, the correctness of which no longer admits, in my mind, of any question. The process is quite elementary, and every step may be verified by any one who is moderately familiar with the principles of Physical Astronomy.
W F Donkin, Note on certain statements in Elementary Works concerning the Specific Heat of Gases, London Edinburgh and Dublin Philosophical Magazine and Journal of Science (4) 28 (191) (1864), 458-461.

A young student of natural science showed me a few days ago the following statement in Galloway's Second Step in Chemistry (London, 1864). It had naturally surprised him, and he asked for an explanation, which I was quite unable to give. (P. 585, paragraph 1321.)
From the calculations of Laplace and Poisson, and the experiments of Clement and Desormes, of De la Roche and Berard, and of Gay-Lussac and Dulong, it has hitherto been assumed ,that the specific heat of a gas under a constant pressure is always greater than the specific heat under a constant volume; but M Regnault has lately found, by an entirely new method, that the difference between the two kinds of specific heat is either null or extremely small.

This paragraph is not accompanied by any note or reference, but it is enclosed in inverted commas, and I soon discovered that it is a translation of a passage in Ganot's Traité élémentaire de Physique (Paris, 1859). See p. 312, end of paragraph 334.

There is an English translation of Ganot, in which the same passage occurs, and is left, as it is in the original, without note or comment.

I applied for further information to some of my scientific friends, and Professor Price pointed out to me a note in Jamin's Cours de Physique, which appeared at first sight to assert the equality of the two kinds of specific heat. The probable explanation of it (vide infra) was suggested to me by Professor H Smith and Sir B Brodie.
W F Donkin, Acoustics. Theoretical (Clarendon Press, Oxford, 1870).

As this is the only portion of a treatise on Acoustics, intended to comprise the practical as well as the theoretical parts of the subject, which will proceed from the pen of its Author, a few words are required to explain the circumstances under which it now appears.

The Author, the late Professor Donkin, has passed away prematurely from the work. It was a work he was peculiarly qualified to undertake, being a mathematician of great attainments and rare taste, and taking an especial interest in the investigation and application of the higher theorems of analysis which are necessary for these subjects. He was, moreover, an accomplished musician, and had a profound theoretical knowledge of the Science of Music.

He began this work early in the year 1867; but he was continually interrupted by severe illness, and was much hindered by the difficulty, and in many instances the impossibility, of obtaining accurate experimental results at the places wherein his delicate health compelled him to spend the winter months of that and the following years. He took, however, so great an interest in the subject, that he continued working at it to within two or three days of his death.

The part now published contains an inquiry into the Vibrations of Strings and Rods, together with an explanation of the more elementary theorems of the subject, and is, in the opinion of its Author, complete in itself; his wish was that it should be published as soon as possible; and he was pleased at knowing that the last pages of it were passing through the Press immediately before the time of his death. It is the first portion of the theoretical part.

It was intended that the second portion should contain the investigation into the Vibrations of Stretched Membranes and Plates; into the Motion of the Molecules of an Elastic Body; and into the Mathematical Theory of Sound. Professor Donkin did not live long enough to complete any part of this section of the work.

The third portion was intended to contain the practical part of the subject; and the theory and practice of Music would have been most fully considered. It is exceedingly to be regretted that the Professor did not live to complete this portion; for the combination of the qualities necessary for it is seldom met with, and he possessed them in a remarkable degree. Not even a sketch or an outline is found amongst his papers. He had formed the plan in his own mind and often talked of it with pleasure. It can now never be written as he would have written it.

Bartholomew Price.

11, St Giles' Oxford,
February 16, 1870.

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