May 26, 192 1] 



NATURE 



m 



known treatise. The tendency in more recent 

 times, however, has been so strong in the direc- 

 tion of physics that less and less space is given in 

 text-books of the calculus to the theory of curves, 

 and the number of students of the theory has 

 probably decreased. But investigation and re- 

 search have, nevertheless, been continuous, and, 

 now that Salmon's treatise is not readily acces- 

 sible, even if it were abreast of modern develop- 

 ments, the need for a good introduction in English 

 to the theory of curves has become clamant ; such 

 an introduction is to be found in Prof. Hilton's 

 book. 



A reader of the book is supposed to possess a 

 knowledge of the more elementary portions of the 

 calculus and of pure and analytical geometry, 

 including the theory of cross-ratio, involution, 

 projection, reciprocation, and inversion. Without 

 a good knowledge of the subjects named the 

 student's progress will not be rapid, and occa- 

 sionally, as, for example, in the study of super- 

 linear branches, some familiarity with the theory 

 of the expansion of algebraic functions is almost 

 a necessity. But any student who is in earnest will 

 find in Prof. Hilton's exposition an excellent guide 

 to the subject of which he treats. The first eight 

 chapters discuss what may be roughly described 

 as the leading principles — singular points, foci, 

 determination of the branches at singular points, 

 and Pliicker's numbers. At an early stage a care- 

 ful treatment of curve-tracing is given, fully illus- 

 trated by well-selected equations, while numerous 

 examples, with hints for the more difficult cases, 

 are provided for practice in this very necessary 

 part of the student's training. 



A compact but careful discussion of the quad- 

 ratic transformation is given in chap. ix. ; to a 

 student new to the subject this discussion should 

 be very illuminating. A good chapter on curves 

 given by a parametric representation is followed 

 by an interesting chapter on "Derived Curves," 

 among which are included evolutes, inverse curves, 

 pedal curves, orthoptic and isoptic loci, cissoids, 

 conchoids, and parallel curves. This chapter is 

 of special interest, as the geometry of the curves 

 considered figures more prominently than in the 

 chapters which discuss the algebraic developments 

 that are necessarily associated with the subject. 

 Later chapters treat chiefly of cubics and quartics, 

 and probably it would be hard to find anywhere 

 a better discussion; the chapters do not always 

 make easy reading, but they are well worth the 

 most careful study. Two excellent chapters on 

 circuits and corresponding ranges and pencils 

 bring the work to a close. 



A valuable feature of the book is the very large 

 NO. 2691, VOL. 107] 



number of examples provided for practice; there 

 can be no better training for the student than 

 the careful study of these examples. Hints for 

 their solution are given in many cases, but the 

 chief advantage is that a student is really intro- 

 duced to the methods of research, and put in a 

 position from which he can undertake independent 

 investigation. 



The book is provided with a good index, but it 

 might be considered, in view of a later edition, 

 whether a special list might not be made of the 

 more important curves, and a connected summary 

 given of their leading geometrical properties. 

 Such summaries as are to be found in the recent 

 work of Brocard and Lemoyne on "Courbes Geo- 

 m^triques Remarquables " are very instructive. 



Aeronautical Treatises. 



(i) Aeronautics in Theory and Experiment. By 

 W. L. Cowley and Dr. H. Levy. Second 

 edition. Pp. xii 4- 331 -f plates. (London: 

 Edward Arnold, 1920.) 255. net. 



(2) A Treatise on Airscrews. By W. E. Park. (The 

 Directly-Useful Technical Series.) Pp. xii-f 308. 

 (London: Chapman and Hall, Ltd., 1920.) 

 215. net. 



(i) " I ^HE second edition of Mr. Cowley and 

 1. Dr. Levy's book is now issued, and 

 the authors have seized the opportunity to modify 

 some of the material. This has become possible 

 by the release of official reports for publication. 

 The new items are of an advanced nature, and 

 the book now contains two sections, "Mathe- 

 matical Theory of Fluid Motion " and " Critical 

 Behaviour of Structures," which are unique in 

 the literature of aeronautics. Both sections are 

 written by the authors as pioneers, for Dr. Levy 

 has a first-hand knowledge of the mathematics 

 of fluid motion and is an original worker in the 

 subject, whilst the "Critical Behaviour of Struc- 

 tures " is the result of joint study by the authors 

 of the complex problems of structural theory. 



Throughout the book there is much more 

 theory than experiment, and for the latter the 

 data are, as usual, taken mainly from the reports 

 of the Advisory Committee for Aeronautics. The 

 selection of items in reference to points under 

 consideration is good, and the book can be re- 

 commended as sound. It is distinctly a student's 

 book, and is not modelled on the needs of the 

 designer like the great bulk of publications on 

 the subject. In range it covers, sometimes in 

 quite an elementary manner, both the aerodynamic 

 and structural problems connected with the aero- 



