NATURE 



[July S, 1909 



THE GEOMETRY OF FORCES. 



Geometric dcr Krcifte. By H. E. Timerding. Pp. 

 xii + 381. (Leipzig-: B. G. Teubner, 190S.) Price 

 16 marks. 



IN this admirable volume Prof. Timerding gives a 

 S3'stematic and original treatment of the geo- 

 metry of forces and force-systems in whidi for the 

 first time, so far as we are aware, an adequate know- 

 ledge of modern geometrical research has been utilised 

 in a text-book of mechanics. 



Ever since the great work of Pliicker, that large 

 and most attractive department of mathematics known 

 as the geometry of the linear complex has been found 

 to be intimately connected with the geometry of forces. 

 It is sufficient to recall the fact that whenever six 

 forces applied to a free body are in equilibrium, the 

 forces must lie respectively on six rays of a linear com- 

 plex. In chapters viii. and ix. of Timerding's book 

 now before us we have an admirable treatment of the 

 application of the theory of the linear complex to the 

 theory of systems of forces. The many interesting 

 matters set forth in these pages show how greatly the 

 advancement both of the geometrical theory and the 

 dynamical theory is promoted by their association. 



The statical and dynamical significance of the linear 

 complex is closely connected with the fact that each 

 ray of the complex is reciprocal to that screw of which 

 the axis is the axis of the complex, while the pitch of 

 the screw is the parameter of the complex. Many of 

 the geometrical properties of the complex follow directly 

 from this general principle. For example, on p. 107 

 it IS shown that four linear complexes have two real 

 or imaginary rays in common. This is an immediate 

 consequence of the fact that one cylindroid can always 

 be found of which every screw is reciprocal to any 

 four given screws. As there are two screws of zero 

 pitch on the cylindroid, these lines are, of course, the 

 two common rays of the four linear complexes defined 

 as being reciprocal to each of the given screws. We 

 congratulate Prof. Timerding on his recognition of 

 the proper place for the linear complex in the fore- 

 front of a text-book on the geometry of forces. 



The theory of screws has received' in this volume a 

 treatment even more ample than that which it has 

 already received in the works of Fiedler, Schell, Budde, 

 Minchin, and more recently in the " Encvclopjidie der 

 mathematischen Wissenschaften." The excellent 

 work of Harry Gravelius, " Theoretische Mechanik 

 Starrer Systeme," contains a complete account of the 

 theory of screws up to the date of its publication in 

 1889. Much of the work done on the subject in the 

 succeeding decade has been available for the " Geo- 

 metric der Krafte." It may, however, be remarked 

 that certain developments of the theory which have 

 appeared since 1900 have not been included in Prof. 

 Timerding's volume. The theory of screw-chains, by 

 which^ the theory of screws has "been extended to any 

 material system, is also not discussed. A suggestive 

 reason for this omission is given in the preface (p. vii), 

 where Prof. Timerding says that, in his opinion, the 

 theory of screw-chains would require a new and 

 voluminous treatment of the whole of mechanics in 

 which the rigid body would appear as the first element. 

 NO, 2071, VOL. 81] 



Observing that the laws for the composition of 

 twists and wrenches are identical, the author, as 

 others have done, uses the word dyname to signify 

 either a twist or a wrench. For a large part of the 

 subject the use of the abstraction signified by the 

 word dyname is very convenient, and considerable 

 use has been made of the important labours of Study 

 on the geometrical theory of dynames. 



In an interesting chapter on " Die Reyeschen 

 Strahlencomplexe " the author brings into its due 

 prominence the fundamental importance of the 

 " Geometric der Lage " in kinematics. This chapter 

 contains many admirable theorems, and we could only 

 wish that such instructive and beautiful ideas as are 

 here set forth were more generallv introduced into 

 the teaching of mechanics. Due acknowledgment is 

 made throughout the work of the important contribu- 

 tions to the geometrical theory of forces by the late 

 Prof. Charles J. Joly. 



The chapter on the cylindroid may be specially com- I 

 mended, and prominence is given to the theorem that 

 the projections of any point on the generators of a 

 cylindroid lie on an ellipse. We may, however, note 

 that the proof here set forth is not that by which the 

 theorem was discovered, as shown in the original 

 volume on the theory of screws published in 1S76. 



A sufficient account is given of the various systems 

 of screw coordinates, and, following the analogv of 

 the resolution of forces. Prof. Timerding uses nota- 

 tion which divides the coordinates of a screw into 

 two groups of three each. It is, however, often con- 

 venient to use the six symmetrical coordinates of .■» 

 screw referred to six co-reciprocal screws. 



We are glad, indeed, to commend this most excel- 

 lent work to the attention of teachers and students of 

 theoretical dynamics. We are sure that if the book 

 were translated into English it would form a very 

 valuable supplement to the existing English books. 

 It would give the student an adequate idea of the 

 extent to which modern geometrical theory and the 

 theory of forces act and react on each other to the vast 

 benefit of both. Robert S. Ball. 



'I 



THE DISTRIBUTION OF GOLD ORES. 



Gold: Its Geological Occurrence 'and Geographical 

 Distribution. By J. Malcolm Maclaren. Pp. 

 xxiii-l-687. (London : The Mining Journal, 1908.) 

 Price 255. net. 



DR. MACLAREN begins his preface with the ' 

 remark that " the writer who would add ■ 

 one more treatise to the literature of the 

 study of ore-deposits must needs show justifi- 

 cation." Any apology for the publication of his 

 useful book is, however, quite unnecessary, for the 

 increase by four times of the gold yield of the world 

 during twenty y'ears has been attended bv a volumin- 

 ous and scattered literature. Students of mining 

 geology will be grateful to any author who undertakes 

 the great labour of compiling a summary of recent 

 work on gold and its distribution. 



The longest and most valuable section of Dr. Mac- 

 laren's book is occupied by an account of the geological 



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