September 30, 1909] 



NA TURE 



39 



:)Vo 



vector, vvhicli can have no scalar part. Again, in 

 Exercise (ij) we have by definition 



£»=i+a + — + — +. . . ' 

 1.2 1.2.3 



where 



a- = a.a, a^ = aa.a, a*=a.aaa, 



and we are to prove that 



t'' = cosa + ai sin a, 

 •where a is the length of a, and a^ is the unit vector 

 along a. Now according to Gibbs a.a- +rt^, so that 

 all the terms of the assumed expansion must be posi- 

 tive. How then can thej' give the sine and cosine? 

 The statements are true only if we use the Hamil- 

 tonian vector whose square is minus the square of its 

 length. The linear vector function is introduced for 

 the discussion of the Uinetics of a rigid body. This is 

 purely Hamiltonian, and is very good so far as it 

 goes. The investigation, however, seems to lock here 

 and there the strength and spontaneity of Tait's clas- 

 sical discussion. 



RETAINING WALLS AND ROAD BRIDGES. 

 (i) Graphical Determination of Earth Slopes, Retain- 

 ing Walls, and Dams. By Prof. C. Prelini. Pp. ix + 

 i2g. (London : A. Constable and Co., Ltd., 1908.) 

 Price 8s. net. 

 (2) The Design of Highway Bridges, and the Calcula- 

 tion of Stresses in Bridge Trusses. By Prof. M. S. 

 Ketchum. Pp. xxi + 544. (New York: The Engin- 

 eering News Publishing Co. ; London : .\. Constable 

 and Co., Ltd., igoS.) Price 16s. net. 

 (i) ' I "HIS book brings together for the use of the 

 -I- engineering student in a handy form for 

 reference the various graphical methods due to Cul- 

 mann, Rebhann, and others, for solving problems 

 connected with earth pressures. 



The first chapter treats of the stability of earth 

 slopes ; the cohesive force in a bank of earth is deter- 

 mined by graphical methods, and hence is deduced the 

 most probable plane of sliding; by means of the 

 parabola of cohesion the various slopes of equilibrium 

 for various heights of bank are determined, and its 

 application to practice is then discussed ; the consider- 

 able economy in excavating deep trenches with slopes 

 correctly designed is proved by worked out examples. 



In the second chapter the design of retaining walls 

 IS taken up; the author points out that all the various 

 theories which have been employed can be divided 

 into two groups, (a) those depending on the theory 

 of the sliding prism, (b) those depending on analytical 

 theory. A graphical solution, due to Rebhann, of the 

 sliding prism type is then given ; this method rs then 

 applied to a series of practical cases, both for retaining 

 walls when surcharged, and when free of surcharge. 

 The variation of pressure with height of wall, and 

 position of the centre of pressure are dealt with, and 

 also the effect of cohesion on the pressure against 

 retaining walls, and the pressure of passive resistance 

 of the earth in the case of abutments which are pushed 

 outwards by arches. 



In the next chapter there is an analytical demon- 

 stration of Rebhann 's theory, and brief statements of 

 NO. 2083, VOL. 81] 



the formulas of Rankine and Weyrauch. In a series 

 of three tables the author gives the values of the earth 

 pressure against walls of different heights as deduced 

 by these three formulae, and the results agree so well 

 that it is evident that any one of the three methods is 

 equally trustworthy from the practical point of view. 



The results obtained in the preceding chapters are 

 applied in chapter iv. to the design of various types 

 of retaining walls ; and the important problem of the 

 determination of the necessary thickness at the base of 

 a retaining wall in order that it may be stable under 

 earth pressure is fully discussed for each type of wall. 

 The last chapter of the book is devoted to masonry 

 dams, which, as the author points out, are simply 

 a particular case of retaining wall, with the material 

 sustained practically frictionless ; it is shown that the 

 most economical profile, theoretically, is a triangular 

 one, but in practice this is an impossible section. 

 The modification needed in order that the dam may 

 have a certain thickness at the top is then discussed, 

 and the pentagonal profile deduced. It is shown that 

 this theoretical profile is the basal form of all modern 

 high dams. The book should prove especially useful 

 to civil engineering students during their final college 

 year. 



(2) While many text-books have been devoted to the 

 design of railway bridges, but little attention has 

 been hitherto given to the equally important question 

 of the design of road bridges, and, although the work 

 of calculating the stresses in the different members 

 is the same for both types, there are, owing to the 

 very different requirements to be met, radical differ- 

 ences in the design of the two classes of bridge. This 

 book, therefore, meets a distinct want, and it will be 

 especially useful to the young designer, in view of 

 the fact that the author has given special attention 

 to the problem of the design of the substructure, which 

 is usually quite neglected in books on bridge design. 

 An entire chapter has been devoted to the design of 

 floor beams, floors, shoes, and pedestals, and other 

 similar details, and it is in regard to such matters 

 that the young engineer most commonly finds the 

 need of help and guidance. 



The ninth chapter will prove of considerable use, not 

 only to the student, but also to the teacher; as the 

 author points out, in order to obtain a thorough 

 knowledge of the calculation of stresses in bridge 

 trusses, it is essential that the student should work 

 through numerous problems— altogether twenty-four 

 problems are worked out in detail in this section of the 

 book, and a second similar one has been added to 

 each of the twenty-four problems as a further exercise 

 for the student to solve with the help afforded by the 

 vvorked-out example; some of the solutions are ob- 

 tained graphically, others by algebraic methods. 

 Another valuable section is that devoted to influence 

 diagrams, or influence lines, which are required in 

 studying the variation of the effect of a moving load 

 or system of loads, on a truss. 



Special attention has been given to the design of 

 high truss steel bridges, and to plate girder bridges, 

 and this section of the book is well illustrated with 

 reproductions of working drawings, which, in spite 



