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LIX. Notices respecting New Boohs. 



The Advanced Part of a Treatise on the Dynamics of a System of 

 Rigid Bodies, being Part II. of a Treatise on the whole subject; with 

 numerous examples. By E. J. Eouth, D.Sc, F.E.S. London : 

 Macmillan and Co. 1884. (Pp. xii+343.) 



THIETT years ago the text-book at Cambridge'on this subject was 

 usually a work which was professedly little more than a Syllabus, 

 consisting of 88 pages of text and 28 pages of examples. It was 

 expected that the student's reading should be supplemented by oral 

 teaching both of College lectures and of the private " coach." Now 

 " the subject is so vast and has so many applications, that a small 

 book can only be made by omitting or treating imperfectly some of 

 its details." These are the words of our author, than whom hardly 

 any one at the present time can speak with greater authority on 

 this branch of Mathematics. 



The general scope of the treatise in this its " fourth edition, 

 revised and enlarged," was indicated by us in our notice of the 

 first part (Phil. Mag. March 1883); and little more is needed than 

 for us to state what portions of the subject Dr. South has handled 

 in this second part. These are discussed under thirteen chapters, 

 respectively headed : — Moving axes and relative motion ; oscilla- 

 tions about equilibrium, and about a state of motion ; motion of a 

 body under the action of no forces and under any forces ; nature 

 of the motion given by linear equations, and the conditions of 

 stability ; free and forced oscillations : determinations of the con- 

 stants of integration in terms of the initial conditions ; applications 

 of the Calculus of Finite Differences, and of the Calculus of Varia- 

 tions ; Precession and Nutation ; motion of a string or chain ; and 

 motion of a membrane. It will thus be seen that there is much 

 new matter; though, on the other hand, there is, of course, much 

 which found a place in the third edition ; but in the last-cited 

 edition we have 564 pages to set against 385 -f 343 pages in this 

 fourth edition. The whole, on perusal, will, however, be found to 

 have been carefully recast, and quite a number of illustrative pro- 

 blems have been added and discussed. The author has not only 

 long worked at his subject, but has been working at it up to the 

 present date. These recent results, which first saw the light in 

 the ' Proceedings of the London Mathematical Society/ have been 

 incorporated here, but in a somewhat different guise ; even results 

 published so lately as in the closing numbers of the Society's 

 present volume of ' Proceedings ' find a place (see p. 243). 



Dr. Eouth has again introduced a few historical notes, which we 

 are always pleased to come across in a text-book, and has removed 

 from an Appendix to a place in the early part (pp. 108, 109) his 

 interesting (" they appear to be new ") properties of the sphero- 

 conic or spherical ellipse. In addition we have some illustrations 

 from the game of billiards (" to supply some results which may be 

 submitted to experiment "), and much light is, by the way, thrown 

 upon (if also derived from) numerous branches of Analysis. The 

 book appears to be very accurately printed ; at any rate there are 



