98 



NA TURE 



[Dec. I, 1887 



to have them established as a foundation from which to 

 start. 



The problem of free vibrations is first treated, and as 

 an example the propagation of plane waves of (i.) normal, 

 (ii.) tangential, displacement is investigated. It is a pity 

 that the author has not here taken the opportunity of 

 illustrating some previous remarks on the discontinuity 

 of the forms of the strain and stress components which 

 necessarily accompanies a change in the nature of the 

 medium, by considering the question of the reflection 

 and refraction of plane waves. 



The general form of the solution of the equations 

 for forced vibrations is next investigated. Then follows 

 the general question of equilibrium. As a simple ex- 

 ample the case of a cylindrical tube under external 

 and internal normal pressures is first treated. It is 

 almost annoying to find the solution of this time- 

 honoured question obtained by starting from the general 

 equations, and whittling them down till the very simple 

 conditions are fulfilled. The equilibrium of a solid sphere, 

 with either surface tractions or displacements given, is 

 treated exactly as in Thomson and Tait's work. The 

 chapter of general solutions closes with an account of 

 Airy's general method for plain stress, with a couple of 

 examples. The printing was unfortunately so far ad- 

 vanced that this had to be left ; though before the book 

 appeared the author had himself shown, in a communi- 

 cation to the London Mathematical Society, that these 

 examples of Airy's are faulty, and that the method applies 

 only to a very limited class of cases. 



Chapter VII. consists mainly of a capital exposition of 

 the solution of St. Venant's problems of the torsion and 

 flexure of prisms. These problems are probably, from a 

 practical point of view, the most important for which an 

 exact solution has been obtained. The author brings 

 out well the bearings of the nature of the solutions on 

 practical questions of construction. The elastic equili. 

 brium and small motions of wires, whether straight or 

 curved, are deduced directly from the results of St. 

 Venant's problems. In connection with this part of the 

 subject, certain interesting questions of stability, due to 

 Mr, Greenhill, are discussed. 



Some cases of the equilibrium and vibrations of plates 

 and shells are considered in Chapter VIII. For the 

 equilibrium of a plate of uniform thickness under a 

 system of surface tractions parallel to its faces and 

 acting on its edges, a solution is obtained by analysis 

 very similar to that used in St. Venant's problem. The 

 case of a thin plate under the action of applied faces 

 satisfying certain conditions is quoted from Thomson 

 and Tait again. 



Two short chapters headed " Impact " and " Viscosity " 

 complete the volume. The former consists of the solu- 

 tion of two problems, one of which, as the author 

 implies, has nothing to do with impulsive change of 

 motion. Indeed, as is well known, the exact treatment 

 of the impact of elastic bodies involves difficulties, even 

 in comparatively simple cases, which have not yet been 

 overcome. In the last chapter the alteration in the form 

 of the dynamical equations is determined, which results 

 from supposing the shearing stress to vary partly as the 

 shear and partly as its rate of change. 



Having thus given some account of the plan of the 



book and the way in which it has been carried out gener- 

 ally, we may offer some remarks on matters of detail. 

 It may be said at once that as regards accuracy there is a 

 good deal to be desired. The table of errata might have 

 been tripled, and would not then have contained all the 

 misprints. In §§ 299, 306, the wholesale omission of 

 signs of summation in the equations makes the analysis^ 

 as given, incorrect ; and there is Kttle doubt but that any- 

 one to whom the matter treated was new would be com- 

 pletely baffled. The inaccuracies, moreover, are not 

 confined to mere misprints. There are one or two positive 

 mistakes in the mathematics. Thus at the bottom of 

 p. 58 it is implied that some condition is necessary in 

 order that a family of surfaces, f {x, y, 2) = ^ (an arbit- 

 rary parameter), may have a system of continuous curves 

 cutting them at right angles ; and in a note at the foot of 

 p. 298 it is stated that, supposing this (entirely imaginary) 

 condition satisfied, two other systems of surfaces can 

 always be found cutting each other and the former sur- 

 faces everywhere at right angles. Now the three para- 

 meters of such a triple system of surfaces have to satisfy 

 three independent partial differential equations, and hence 

 no one of the three can be taken arbitrarily. Statements 

 and reasoning are, in several passages, founded on this 

 erroneous conception. Closely allied with this is the 

 construction given in § 216 for tubes of stress. It is here 

 practically assumed that a given continuous system of 

 curves can always be cut at right angles by a family of 

 continuous surfaces. 



An appendix at the end of Chapter II., on "The 

 Geometry of Strains," might have been omitted v/ith 

 advantage. It has no very obvious connection with the 

 preceding chapter, but is devoted to an apparently new 

 classification of vector quantities, in which a velocity and 

 a force are the types of the one group, while an angular 

 and a couple are those of the other ! Again, in §§ 270, 

 271, the solution of a physical problem is made to appear 

 to depend on the choice of an origin. The question 

 treated is the free normal vibrations of a plate; and, 

 after using d and -d' to denote the abscissae of the 

 two faces, and making the result appear to depend on d/d', 

 the question is simplified by taking the origin midway 

 between the faces. Indeed, frequently throughout the 

 book one is reminded of Clerk Maxwell's remark on " the 

 state of a mind conscious of knowing the absolute position 

 of a point." 



These slips, such as they are, and an occasional 

 obscurity of language, are but slight blemishes on a valu- 

 able book. A friendly but independent criticism of the 

 proof-sheets while the book was passing through the press 

 might have removed them all, and no doubt will in a new 

 edition. 



The figures throughout are excellent. 



THE VOLCANIC AND CORAL ISLANDS OF 

 THE SOLOMON GROUP. 



The Solotnon Islands: their Geology, General Features, \ 

 and Suitability for Colonization. By H. B. Guppy, ' 

 M.B., F.G.S., late Surgeon R.N. (London: Swan 

 Sonnenschein, Lowrey, and Co., 1887.) 



SURGEONS in Her Majesty's navy are favoured 

 beyond most men in the possession of abundant 

 leisure and freedom from many of the common cares of 



