5«2 



NATURE 



[February 29, 19 12 



the jjoneral scope of the book. It is probable that 

 the averaR^e student would do better to follow a course 

 of this kind, and then single out some subject for 

 specialised studv rather than to spend time on repro- 

 ducinfif analytical investigations about the truth of 

 which no doubt exists. 



We notice that both Weber and Lamb assume that 

 in a membrane the stress is the same in all directions 

 — in other words, isotropic. Is this necessary? 

 Analytically speaking, if the stress is homogeneous 

 but not isotropic the membrane can be projected 

 orthogonally into one in isotropic stress ; on the other 

 hand, it is fairly certain that in an actual membrane, 

 such as that of a concert drum, it is very difficult 

 to adjust the tension round the boundary so as to 

 make the stresses either homogeneous or isotropic. 

 A material membrane probably differs in its physical 

 properties from the ideal membrane (Lamb, p. 189) 

 to the same extent that the substance of which it is 

 composed differs from a fluid. If this view is not 

 correct further discussion is needed. 



(6) Reference to notation under (4) leads us to Mr. 

 Robert Innes's short Cape Town note, in which he 

 recommends the use of small letters, Roman, italic, 

 Clarendon, and Greek for symbols of quantity, and 

 capital letters for symbols of operations, such as S for 

 sine, D for differential coefficient. He does not tell 

 us what he would do with separate differentials. In 

 view of the fact that after these many years we still 

 have to write cosec x for what is logically sin-'x, 

 and that a Frenchman or German cannot say such a 

 number as 394 in the form in which it is written, it is 

 not much use suggesting- reforms, except in notation 

 the use of which is very limited. 



(7) In a review like the present, it would be impos- 

 sible to enter into a detailed criticism of the German 

 edition of Dr. Oskar Bolza's "Lectures on the Cal- 

 culus of Variations." While based on his American 

 work bearing the same title, these three volumes aim 

 at a more comprehensive treatment of the subject ; at 

 the same time, the author does not claim to have 

 exhausted the theory even in the seven hundred 

 pages which he has devoted to it. W'hat he has 

 rather aimed at has been to give a fairly elementarv 

 outline of the main principles of the calculus of 

 variations, together with a more detailed treatment 

 of its modern developments. The author's claim to 

 have clearly expounded the fundamental definitions 

 and methods, with the assistance of suitable geo- 

 metrical and other illustrations, is fully justified by a 

 survey of the contents. At present there are only a 

 few people in England who study calculus of varia- 

 tions, and it is to be hoped that the subject will 

 become more popular in the future in view of its 

 important applications to physical problems. "Differ- 

 entiation of an integral," which leads directly to 

 " variation of an integral," is really one of the easiest 

 things in the calculus to treat in an elementan.- wav, 

 and there can be no excuse for keeping this study, 

 so to speak, under lock and key, only to be shown 

 to students on rare occasions. The student of phvsics 

 who goes no further than the first chapter of this 

 book will be able to obtain a proof of Lagrange's 

 equations of motion, together with a knowledge of 



NO.- 2209, VOL. 88] 



the principle of least action, which will do him f.ir 

 more good than letting x equal a function of /, 9, and 

 ^, and writing out by heart a proof of these equations 

 involving some juggling with differential coefficients 

 the physical interpretation of which is not obvious. 



On the other hand, the need for an exposition of 

 modern developments of the subject is shown by the 

 many recent papers that have appeared. For example, 

 a paper in German on the invariant form u{ 

 the second variation of a double integral is contributed 

 to the Proceedings of the Tokyo Mathematical and 

 Physical Society for September, 191 1, by M. Fuji- 

 wara. 



(8) Closely allied, as leading to a generalised form 

 of the principle of least action, are Prof. Konir 

 berger's papers on the principles of mechanics for < 



or more dependent variables, of which the sec 

 part has reached us. 



(9) .\n examination of the contents of "Theoretic 

 Mechanik " leads to the impression tliat Dr. Tin 

 ding showed want of judgment in undertaking 

 translation into German of a book like that of Mai > 

 longo's. It is known that Profs. Marcolongo .d ! 

 Burali Forti have been engaged in drawing up i 

 report on vector notations, as to the value of which 

 considerable differences of opinion exist. We should 

 not, however, take any exception to the book on this 

 ground; but when we find in the second chaj • 

 difficult theorems in potential analysis, such as Stok- - 

 and Green's theorems, and Poinsot's construction Ux 

 the motion of a rigid body under no forces in 

 chapter vii., while it is not until much later on that 

 the author deals with such elementary notions as t' - 

 parallelogram of forces, the principle of the lev 

 equilibrium on an inclined plane, and the principL 

 Archimedes, the book may fairly be regarded as aff( 

 ing a lesson as to how mechanics should not be 

 taught. It forms a striking contrast to the clear and 

 practical exposition of advanced dynamics contained 

 in Webster's book, published by the same firm. It is 

 becoming more and more recognised every day that 

 the study of mechanics should be approached by be- ; 

 ginners from the experimental side, and for this pur- ' 

 pose elementary statics and Tiydrostatics form the best ; 

 starting point. With such a preparation, a pupil may ' 

 be able in time to grasp the nature and use of vector 'k 

 analysis, but to start him, as this book does, at the i 

 wrong end would be fatal. ■ 



(10) In his paper on curvature, Prof. Cailler 

 has set before himself the task of formulating a ' 

 theory of curvature of a sufficiently comprehensi' ■■ 

 character to cover the two cases of Riemann's ; 

 Lobatschewski's non-Euclidean geometries. To 



this he has found it necessary to start with an al^ 

 braic definition of curvature based on considerations, 

 of kinematic geometry, and while admitting that some | 

 of the notions in his paper are of a somewhat abstract 

 character. Prof. Cailler claims that the nature of the 

 problem and the differences existing between the 

 various kinds of space to which it refers render such a 

 treatment necessary if it is to possess the requir-.d 

 degree of generality. 



(ii) The volume of the London Proceedings before 

 us continues to afford evidence of the good work that 



