244 



NA TURE 



[August 19, 1922 



service by including them in his books, even if only in 

 the form of an afterthought. 



(2) Prof. Love's " Theoretical Mechanics " is a book 

 that serious students of dynamics cannot be without : 

 the discussions of the principles are illuminating, and 

 the collections of examples are useful to both teacher 

 and pupil. Only a few changes have been made in the 

 recently issued third edition. Perhaps it is permissible 

 to suggest that the book would be immensely more 

 useful if it partook of the nature of a text-book, and 

 included a much larger number of worked examples. 

 The student's main difficulty in dynamics is not in 

 learning the comparatively restricted number of ideas 

 and methods given in the usual honours courses, but 

 rather in obtaining the necessary experience for using 

 these ideas and methods successfully in the problems 

 presented by nature. By far the most effective help 

 that can be given him is that contained in a judiciously 

 selected and carefully graduated series of worked 

 problems, whe»e the efficiency value of each process is 

 made evident. 



As the author emphasises the importance of the 

 fundamental principles, the volume would be the 

 right place for a brief account of relativity in dynamics. 

 To leave this latest phase of modern scientific reform 

 to the physicist and the philosopher is a mistake that 

 applied mathematicians should endeavour to counteract. 



Publishers no doubt know their business and do not 

 need the advice of academic men. A protest must 

 nevertheless be raised against excessive prices. The 

 price of this new edition will prevent its sale among 

 just those young students whose mechanical ideas the 

 author wishes to influence. 



(3) and (4) The study of two-dimensional problems is 

 of great interest in several branches of applied mathe- 

 matics, as, for instance, in potential theory, electricity, 

 and hydrodynamics. It often happens that when a 

 three-dimensional problem of importance cannot be 

 solved, the two-dimensional case is amenable to modern 

 mathematical methods and its solution sheds much 

 light on the general problem. This has been the case 

 particularly in hydrodynamics. 



The present volumes are the first two parts of a 

 treatise on two-dimensional hydrodynamics. Part I. 

 gives the theory of the complex variable and conformal 

 representation, which is followed by a statement of 

 the equations of motion of a fluid in two dimensions. 

 Problems with boundaries consisting of free stream 

 lines only, and with boundaries consisting of fixed 

 barriers only, are then discussed. Part II. deals with 

 jets and other problems, involving both fixed and free 

 boundaries, while Part III. will deal with wave-motion. 



Of the different types of problems discussed by 

 Prof. Cisotti, perhaps the most interesting is that of 

 NO. 2755, VOL. I io] 



discontinuous motion of fluid past a fixed barrier — a 

 problem that has some bearing on the modern subject 

 of aerodynamics. When the barrier is plane, and the 

 motion is assumed to be irrotational, with free stream 

 lines, the problem has been solved by the use of what 

 constitutes one of the most elegant processes of mathe- 

 matical reasoning. Curved barriers, however, have 

 so far defied solution, except in the sense that when a 

 solution is suggested one can obtain equations which 

 define the barrier appropriate to the solution. The 

 problem of the curved barrier may almost be described 

 as one of the classical problems of hydrodynamics. 

 Several interesting cases have been discussed, in par- 

 ticular by Prof. Cisotti himself, Villat, and others. 



The ordinary text-book process of solving two-dimen- 

 sional problems in hydrodynamics is to seek a relation 

 between the complex variable that represents the 

 geometry of the actual motion and the complex variable 

 involving the velocity potential and the stream-line 

 function. An intermediary variable, which is essen- 

 tially representative of the velocity vector, is often 

 useful. In dealing with discontinuous motion past 

 barriers consisting of plane surfaces, a further inter- 

 mediary variable is needed, based on the Schwartz- 

 Christoffel transformation : the problem is then 

 reduced to quadratures. 



For curved barriers, however, this is insufficient, 

 and a new type of transformation has been found 

 necessary. The essential idea of this transformation 

 is to make the barrier correspond to a semicircle in a 

 new Argand diagram. The general solution of the 

 problem is then defined in terms of a Taylor expansion, 

 and the choice of the coefficients in this expansion 

 determines any particular curved barrier. Elegant 

 formulae exist for finding the pressure components on 

 the barrier, and the line of action of the resultant 

 pressure, but an explicit statement for the latter has 

 not yet been published. 



This process, due to Levi-Civita and others, can be 

 made to yield numerical results of considerable interest. 

 Brillouin has given the working for a set of barriers 

 defined by a certain choice of the coefficients in the 

 above-mentioned series. Further, by a process of 

 approximation, circular and elliptic barriers admit of 

 numerical solution. 



Prof. Cisotti's resume of the progress in this problem 

 during the last fifteen years is masterly, and of great 

 use to researchers in the subject. It seems, however, 

 that the footnote on p. 179 is based on a misapprehen- 

 sion. Brillouin has given the conditions that must be 

 satisfied if the free stream lines are to have finite 

 curvature where they leave the barrier. The author 

 urges that these conditions are not necessary. He is 

 right, but Brillouin does not mean that these conditions 



