May 3, 1900] 



NATURE 



mentioned. Another appendix, on the zoological material, 

 is principally devoted to the description of a new arachnid 

 and of a new oligochicte annelid collected on the snow. 

 An appendix on the rocks and minerals is for the most 

 part a discussion of Russell's previous work, but contams 

 the information that the outcrops near the summit of the 

 mountain consist of typical diorite passing locally into 

 hornblendite. 



The ascent of Mount St. Elias was an achievement 

 worthy of a prince, and this handsome volume is worthy 

 of the achievement. Beautifully printed, magnificently 

 illustrated and tastefully bound, it reflects credit upon all 

 concerned in its production. But (alas ! the inevitable 

 but !) it has no index. G. W. L. 



A HYDRODYNAMICAL THEORY OF ACTION 



AT A DISTANCE. 

 Vorlestingcfi iiber hydrodynamische Fernkrdjte nach 



C. A. Bjcrknes' Theorie. Von V. Bjerknes. Band i. 



Pp. 338 ; with 40 figures. (Leipzig : Johann Ambrosius 



Barth, 1900.) 



THEORIES of matter — or should we not rather call 

 them theories of /^r^^, since, in "explaining" the 

 properties of matter, we are mainly concerned with those 

 manifestations which we say are due to "force" — natur- 

 ally fall into two distinct classes. The first class includes 

 those hypotheses which regard continuous matter as 

 being built up of discrete particles, and the direct action 

 of finite portions of matter as being due to action 

 at a distance of these particles. The second class in- 

 cludes those hypotheses which regard these particles as 

 singularities in a continuous medium, and which attri- 

 bute their action at a distance to the direct agency of 

 the medium. In a certain sense, these two theories are 

 reciprocal. In both, certain attributes are localised at 

 points, and it is necessary to bridge over the distance 

 between these points. According to the first hypo- 

 thesis, a field of force pervades the intervening gaps ; 

 according to the second, they are filled with a dis- 

 tribution of mass. The belief that both hypotheses are 

 possible, enables us to imagine that there may be no 

 limit to the smallness of the scale on which Nature con- 

 ducts her operations, the phenomena occurring in any 

 region being made to depend in their turn on others oc- 

 <;urring in the far more minute regions which are regarded 

 as constituting its ultimate elements, and these elements 

 being in turn capable of further subdivision, and so on 

 indefinitely. 



In 1852, Lejeune-Dirichlet, being unacquainted with 

 the works of Green and Stokes on this subject, published 

 a paper containing the solution of the problem of the 

 motion of a sphere in an incompressible fluid. In a 

 course of lectures given at Gottingen in 1855-56, Dirich- 

 let gave the corresponding solution for a sphere fixed in 

 a steady current, and invited his pupils to attempt the 

 -^olution for an ellipsoid. Among these pupils were 

 -cohering, who solved the problem, and C. A. Bjerknes, 

 who gave a generalisation for space of n dimensions. At 

 this time the doctrine of action at a distance may have 

 been said to be at its zenith, and Gottingen had given 

 birth only a few years previously to the last brilliant pro- 

 duct of that doctrine, Weber's Law. As a foreigner, 

 NO. 1592. VOL. 62] 



Bjerknes was, however, less influenced by the views then 

 prevailing in the Gottingen school, and a volume of 

 Euler's correspondence falling into his hands caused him 

 to oppose the doctrine of action at a distance. A fresh 

 light was thrown on the hypothesis of a continuous all- 

 pervading medium by Dirichlet's discovery that a sphere 

 moving in an incompressible perfect fluid experiences no 

 retardation from the fluid, and an impetus was given to 

 Bjerknes to develop Dirichlet's investigations in a 

 direction widely differing from anything then contem- 

 plated by his professor. 



From the effects of purely translational motions of two 

 spheres, Bjerknes was led on to consider the mutual 

 actions of two pulsating spheres, and discovered that 

 such spheres attract or repel one another according 

 as their phases are the same or opposite, the law of 

 force being that of the inverse square. Bjerknes found, 

 moreover, that the expressions for the forces acting 

 on a sphere moving in liquid consisted of two terms, 

 which he distinguished as " inductional forces " and 

 " energy forces," a result which he arrived at by con- 

 sidering the expressions for the pressures on the spheres, 

 but which might have been found more readily had 

 Thomson and Tail's application of Hamilton's principle 

 been then known to him. About 1875, Bjerknes pub- 

 lished a paper in which he established the hydro- 

 dynamical law of action and reaction, and the analogy 

 with electric and magnetic action at a distance ; and in 

 the following year he gave an independent investigation 

 based on the Hamiltonian principle. 



From 1875 onwards, Bjerknes appears to have occu- 

 pied himself chiefly with the terms of lowest order in the 

 expressions for the forces ; and in 1878 he discovered the 

 law of rotation for oscillating spheres. Since then he 

 seems to have devoted his attention mainly to electric and 

 magnetic analogies, and in the middle of his eightieth 

 year he completed the discussion of the " inductional 

 forces," and by this means pushed the analogy between 

 hydrodynamic action at a distance and electromagnetic 

 phenomena as far as it could be pushed without depart- 

 ing from the fundamental hypotheses. 



A complete account of these investigations was never 

 published, and it remained for his son. Prof. V. Bjerknes, 

 to embody them in the present volume. For three years 

 Prof. V. Bjerknes has delivered courses of lectures on 

 the subject at the University of Stockholm, and the book 

 is practically based on these lectures. It is divided into 

 four parts : the first, an introductory part, dealing with 

 the general principles of vector fields and hydro- 

 dynamical equations ; the second, dealing with the motion 

 of the liquid surrounding a system of moving spheres 

 treated from a kinematical standpoint ; the third, dealing 

 with the influence of the pressures on the motion of the 

 spheres themselves ; and the fourth, with the theory of 

 apparent actions at a distance, of hydrodynamical origin. 

 In the second part, the diagrams of the stream lines due 

 to a moving, oscillating or pulsating sphere in various 

 fields of force are noticeable for their elegance. 



It is to be wished that the courses of lectures which 

 Prof. V. Bjerknes delivered on the work of his father 

 could be taken as models of what university lectures 

 should be, for the development of a theory such as the 

 present affords an excellent and not difficult insight into 



