# Piers Bohl's publications

We list below all the papers written by Piers Bohl and also give a couple of books containing collections of his papers. We give brief comments of the contents of some of the papers.

- Piers Bohl, Das Gesetz der molecularen Attraction,
*Annalen der Physik und Chemie, Neue Folge***XXXVI**(1887), 334-346.

- Piers Bohl, Ueber eine Verallgemeinerung des dritten Kepler'schen Gesetzes,
*(Schlömilch's) Zeitschrift für Mathematik und Physik***XXXV**(1890), 188-191.

- Piers Bohl,
*Ueber die Darstellung von Functionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten*(Jurjew (Dorpat): E J Karow, 1893), 31 pages.

In this dissertation Bohl established a necessary and sufficient condition for the representability of a function by a series of the type described in the title of the dissertation.

- Piers Bohl,
*Ueber einige Differentialgleichungen allgemeinen Charakters, welche in der Mechanik anwendbar sind*(Dorpat, 1900), 113 pages.

Here the results of Bohl's Master's Thesis (equivalent to a Ph.D.) Ueber die Darstellung von Functionen einer Variable durch trigonometrische Reihen mit mehreren einer Variable proportionalen Argumenten (1893) are used to determine the possibility of trigonometric solutions of certain differential equations of general character.

- Piers Bohl, Über die Bewegung eines mechanischen Systems in der Nähe einer Gleichgewichtslage,
*Journal für die Reine und Angewandte Mathematik***127**(1904), 179-276.

This paper examines the movement of a mechanical system near an equilibrium position given by Lagrange's equations. There are three distinct cases: 1. There are only stable coordinates. A well-known sentence by Dirichlet does the job. 2. There are only unstable coordinates. 3. There are stable and unstable coordinates. The investigation of this case forms the subject of this paper.

- Piers Bohl, Über eine Differentialgleichung der Störungstheorie,
*Journal für die Reine und Angewandte Mathematik***131**(1906), 268-321.

The present treatise continues an investigation of one already made by Lindstedt (*Beitrag zur Integration der Differentialgleichungen der Störungstheorie*, 1883) and by Poincaré (*Les méthodes nouvelles de la mécanique céleste*, 1893) treating the differential equations from celestial mechanics with the help of theorems which the author Bohl derived in his dissertation (Dorpat 1893).

- Piers Bohl, Über ein Dreikörperproblem,
*(Schlömilch's) Zeitschrift für Mathematik und Physik***54**(1907), 381-418.

In this work on the 3-body problem, the three bodies are assumed to be rigid. Two are spheres (Saturn and a satellite), the third is a ring (around Saturn). Through a series of theorems, which refer almost exclusively to inequalities, the following theorem is proved in an extremely ingenious manner: It is possible to measure the dimensions and masses of the three bodies, as well as the initial position and initial movement, with its inner edge touching Saturn. However, it is also possible to choose the initial speed of the centre of gravity of the ring so that there is no collision between Saturn and the ring at all, so that the movement continues without end.

- Piers Bohl, Zur Theorie der trinomischen Gleichungen,
*Mathematische Annalen***65**(1908), 556-566.

A problem occurring in the theory of secular perturbations led Bohl to a rule to determine the number of roots of a trinomial equation whose absolute value is less than an arbitrarily given positive quantity.

- Piers Bohl, Über ein in der Theorie der säkularen Störungen vorkommendes Problem,
*Journal für die Reine und Angewandte Mathematik***135**(1909), 189-283.

- Piers Bohl, Sur certaines équations différentielles d'un type général utilisables en mécanique (Translated from Russian by Mlle Tarnarider),
*Bulletin de la Société Mathématique de France***38**(1910), 5-138.

French translation of Bohl's doctoral thesis (D.Sc. standard required).

- Piers Bohl, Bemerkungen zur Theorie der säkularen Störungen,
*Mathematische Annalen***73**(1912), 295-296.

Brief comments on Felix Berstein's paper Über eine Anwendung der Mengenlehre (1911) which is turn makes reference to Bohl's paper [9]*Über ein in der Theorie der säkularen Störungen vorkommendes Problem*.

- Piers Bohl, Über Differentialgleichungen,
*Journal für die Reine und Angewandte Mathematik***144**(1914), 284-313.

These investigations of theoretical mechanics are based on certain statements gained through experience, reasoning and guesswork, which, however, do not claim to be accurate, but are approximate. On the one hand, there is the task of examining the movement in question on the basis of the available data; on the other hand, it is essential to assess those deviations which are to be expected in the results obtained due to the inaccuracy mentioned. The first task is usually associated with solving ordinary differential equations. This solution only happens exceptionally through integration "in finite form". Usually one is dependent on approximation methods, of which the "method of successive approximations" is particularly worth mentioning. Regarding the second task, one will often, though not always, be inclined to take the approximate nature of the present data into account by supplementing the differential equations with summands, which are only subject to the condition that they become absolutely not greater than certain constants. In this case it is natural not to speak of differential equations but of*Differential equations*. - If you restrict yourself to a predefined finite time interval, analysis already has methods for examining the questions mentioned in cases of considerable generality. However, it is of interest to find classes of differential inequalities or differential equations in which the mentioned limitation can be avoided, so that forms of movement that are unlimited in terms of time are subject to consideration.

- Piers Bohl, Über die hinsichtlich der unabhängigen und abhängigen Variabeln periodische Differentialgleichung erster Ordnung,
*Acta Mathematica***40**(1916), 321-336.

- A D Myskis and I M Rabinovic (eds.),
*P G Bohl, Selected Works*(Russian) (Riga, 1961).

Contains Russian versions of the papers [2], [10], [5], [6], [9], [11], [12], [13]. Also has an introductory article on the life and work of Piers Bohl as well as a portrait.

- Piers Bohl, Darstellung und Anwendung der Invarianten der linearen Differentialgleichungen (Russian), Latvi-ski- Matematicheski- Ezhegodnik 12 (1973), 237-268.

This is the first publication (translated from German to Russian by I Rabinovics) of Piers Bohl's manuscript, presented in 1886 to the competition of the scientific works of students of Dorpat University.

- L Reizins (ed.),
*P Bohl Collected Works*(Russian) (Verlag Zinatne, Riga, 1974), 517 pages.

This book is a collection of the scientific works of Piers Bohl during his lifetime. The book consists of an introduction and two parts divided into chapters. In the final part of the book six works are presented, in which problems from mathematics and mechanics are developed. In the beginning there is the important work on the theory of invariant transformations and the application of this theory to linear differential equations. The next two works concern Kepler's third law and the development of a function of many variables in a trigonometric series. The first part is on differential equations of a general character which are applied to mechanics and this part is divided into four chapters. The third and fourth chapters are very interesting, here two methods are given for solving differential equations and applications to mechanical problems are given. The second part concerns the motion of a mechanical system in the neighbourhood of the position of equilibrium. In the second chapter a theorem is given on the existence of solutions and on the conditions for asymptotic stability. Of the six works in the final part of the book, the one on the three-body problem is particularly interesting.

Last Updated September 2020