NATURE 



73 



THURSDAY, MAY 26, 1892. 



MATHEMATICS USED IN PHYSICS, 

 Einleitung in die Theoretische Physik. Von Victor von 

 Lange. Second Edition, Enlarged and Revised. 

 (Braunschweig: Vieweg, 1891.) 



THIS work is intended to give an account of the 

 mathematical processes employed in physical in- 

 vestigations. It is divided into chapters dealing with the 

 various branches of physics, mechanics, gravitation, 

 magnetism, electricity, solids, fluids, gases, light, and 

 heat. It is very difficult in such a book to decide how 

 far to go in mathematical processes, and Herr von Lange 

 has exercised his discretion wisely in this matter. At the 

 other limit of how little to assume known he has certainly 

 not erred in the direction of assuming too much, for he 

 introduces proofs of simple differentiations and integra- 

 tions when he requires them, which had much better be 

 learnt continuously in an elementary treatise on the cal- 

 culus. No English student would use a book of this 

 advanced character without some preliminary mathe- 

 matical training, and it is very doubtful whether anybody 

 picking up the calculus in this haphazard fashion could 

 ever use it in his own investigations ; and if it is no use 

 to him for this, would it not be a great saving of time 

 and energy for him to depend on the investigations of 

 others without going through all their work, just as an 

 investigator of magnetic declination need hardly expect 

 to have time to work through the lunar and planetary 

 theories that help in the calculations of the Nautical 

 Almanac he uses ? A work of this kind is of great service 

 as a concentrated store of information for those who want 

 to study physics, and who have sufficient mathematical 

 ability and training to be able to use the mathematical 

 processes involved ; but it cannot successfully compete 

 with special treatises on the elements of solid geometry, 

 differential calculus, &c., as a means of supplying the 

 mathematical training required in order to use these 

 processes. 



Some readers may be disposed to doubt whether it is 

 worthwhile introducing into a work of the scope of thisbook 

 any elementary dynamics. The subject, however, wastes 

 only a few pages, and it may very well be worth while in- 

 troducing it in order to avoid references and explanations 

 that might be quite as long. His discussion of the 

 nature of mass is hardly satisfactory without a descrip- 

 tion of apparatus and methods of experimenting, but, so 

 far as it goes, is fairly sound. He does not point out 

 with sufficient clearness where definition ends and ob- 

 servation comes in. These are, however, really physical 

 questions, with which a mathematical work might very 

 well dispense. In discussing the rotation of a solid sub- 

 ject to forces, he bases his investigation on Airy's 

 mathematical tracts, but he does not safeguard himself 

 with all the provisos Airy so carefully introduces ; and in 

 consequence there are many pitfalls, carefully hidden. 

 The method is based upon supposing the body given 

 a series of blows, and appears on the face of it to be purely 

 kinematical. It is on the other hand evident that, in 

 general, dynamical questions, such as the centrifugal ac- 

 NO 1178, VOL. 46] 



celeration introduced when the ax^s 1 f rotation is not a 

 principal one, must come into consideration when dis- 

 cussing the forces that must be applied to a real body in 

 order to make it move in a given way. A student of this 

 investigation would be puzzled to understand how it 

 happens that a solid sphere, when rotating round an axis 

 and given a blow, begins to rotate round a new axis, new 

 both inside the sphere, and in space, while a gyroscope 

 takes up a wobble. It is possible by a series of blows 

 given to a sphere to cause its axis of rotation to move 

 round in space while preserving its position in the sphere, 

 but a series of blows in general would not produce this 

 result. The kinematic investigation of rotation of a soHd 

 round an axis accompanied by an angular acceleration 

 round a rectangular axis is an interesting geometrical 

 question, but must be carefully distinguished from the 

 dynamical question of what forces must be applied to a 

 real solid in order to produce this motion, and these two 

 different questions not being sufficiently clearly distin- 

 guished make the investigation unsatisfactory. In con- 

 nection with the motion of a solid, it is to be regretted 

 that a short account of the theory of screws was not 

 included. 



Under gravitation at one place, there is a full account 

 of free fall, pendulums, balances, bifilar suspensions, 

 torsion balance, &c. Then he proceeds to questions 

 depending on gravitation at different places, the figure of 

 the earth, the constant of gravitation. Here he mentions 

 Foucault's pendulum, and notices that the elementary 

 investigation is insufficient, without, however, giving 

 more than the result of the complete investigation, not 

 even explaining why the elementary investigation fails, 

 owing to the precessional motion of the axes of the 

 ellipse in which the bob of the pendulum necessarily 

 moves, and which becomes comparable with the motion 

 looked for, unless the amplitude be very small and the 

 suspending thread very long. This chapter concludes 

 with an account of the theorems connected with forces 

 varying inversely as the square of the distance. It is 

 doubtful whether it would not have been better to deal 

 with this subject in the first place from the hydrodynamical 

 point of view. Such theorems as that the flow is equal 

 across every section of a tube of flow, and its numerous 

 consequences, such as that equal quantities of electricity 

 exist at the ends of a tube of force, that the total normal 

 force over any surface is equal to 47r times the quantity of 

 electricity within, &c., are all intuitively evident in hydro- 

 dynamics, and it is well to call a student's attention to 

 the way in which he can safely argue from the familiar to 

 the unfamiliar. 



The chapter on magnetism is very complete, though 

 the action of two magnets on one another is done in 

 a fearfully long-winded way ; and in the account of the 

 determination of magnetic declination the spherical 

 trigonometry required in order to calculate the azimuth 

 of the terrestrial meridian from the astronomical obser- 

 vations is not given. It would also appear as if the 

 determination of variations of dip by means of an in- 

 duction vertical force magnetometer were quite a different 

 thing from determinations of the variation of vertical 

 force by means of a balance magnetometer. Magnetic 

 induction is the usual mathematical investigation of simple 

 cases where the permeability is assumed constant. An 



