June 8, 1916] 



NATURE 



299 



memoirs are conveniently arranged at the ends of 

 the chapters. 



Part i. is concerned with the processes of 

 iinalysis. After an introduction to complex 

 numbers, continuous functions, and the more 

 fundamental theory of convergence and uni- 

 formity, the reader passes to the theory of 

 Riemann integration. Analytic functions are then 

 introduced, and an account of Cauchy's theory of 

 residues is followed by the theory of the develop- 

 ment of functions in various forms of infinite 

 series. The chapter on asymptotic expansions and 

 summable series is very compact, and in the 

 ensuing chapter, on Fourier series, the authors 

 have taken the bold course of treating these series 

 by the elegant means of Ces^ro's theory of sum- 

 mable series, instead of by Dirichlet's method. 

 But as the theory of these series only appeals to 

 the pure mathematician, who finds Dirichlet's 

 method equally difficult, this course appears to be 

 justified. Part i. concludes with a valuable 

 chapter on integral equations, which, like those 

 on the theory of integration and linear differential 

 equations, is new. 



Part ii. is devoted to the theory of the special 

 transcendental functions, and commences with a 

 very complete account of the Gamma function. 

 The statement that this function was defined by 

 £uler as an integral is slightly misleading; he 

 obtained a limit of a product by interpolation from 

 factorials, proved it equal to a Beta integral, and 

 thence derived the Gamma integral. Possibly, 

 however, the writers do not regard the product 

 as being suitable for a definition, for it is not of 

 NVeierstrass's canonical form, and, indeed, it is 

 difficult to show that it represents an analytic 

 function. A sketch of the theory of the Zeta 

 function of Riemann, from the point of view of 

 analytic functions, is given, although an account 

 of its applications to prime numbers seems to have 

 been considered beyond the scope of the book. 

 The work of Mellin and Barnes, which has ap- 

 peared since the first edition, has enabled the 

 authors to give a more brief and systematic 

 account of the hypergeometric function and of its 

 "confluent" form. As particular cases of these 

 functions, the harmonics of the parabolic cylinder, 

 and, of course, the Bessel functions, are con- 

 sidered in some detail. A chapter is devoted to 

 the differential equations of mathematical physics, 

 and a pleasing novelty is introduced into their 

 treatment. The authors are successful in reducing 

 to a minimum the labour inherent in a discussion 

 of Mathieu's elliptic cylinder functions. The book 

 concludes with three long and interesting chapters 

 on elliptic functions, and it is pleasant to observe 

 that Jacobi's notation for the Theta functions has 

 been retained on account of its historical interest. 

 Moreover, it is actually the most convenient of 

 those in existence. 



In matters of general arrangement the book is 

 excellent throughout. Peano's system of para- 

 graphing is adopted, and the reviewer can only 

 express the hope that the system may become more 

 universal. An appendix gives the essentials of 

 the more elementary theory of simpler functions, 

 NO. 2432, VOL. 97] 



and the index is noticeably complete. The ex- 

 amples are numerous and well selected from the 

 point of view of the student who wishes to pursue 

 the subject. But perhaps the most characteristic 

 feature of the book is its success in giving rigorous 

 proofs of theorems without relapsing into the 

 dullness too often associated with rigour. In 

 every respect it is worthy of the traditions of the 

 Cambridge University Press. 



DOCILITY AND OTHER DISEASES. 

 (i) The Nemesis of Docility: A Study of German 



Character. By E. Holmes. Pp. vii + 264. 



(London: Constable and Co., Ltd., 1916-) 



Price 45. 6d. net. 

 (2) La Guerre et la Pensee Medicale. By Prof. 



Ricardo Jorge. Pp. 63. (Lisbon, 1916.) 



(i) P^Y docility the author means "readiness to 

 -L' obey for the sake of obeying, avidity for 

 commands and instructions, reluctance to accept re- 

 sponsibility or exercise initiative, inability to react 

 against the pressure of autocratic authority"; 

 and this is what is wrong with Germany, where 

 a slavishly docile majority is as wax in the hands 

 of a dogmatic and domineering minority. The 

 Germans lost their early domestic freedom in be- 

 coming feudalised, and they failed to recover it 

 because of the disruptive influences of tribalism. 

 The ultra-docility has grown and is obvious to- 

 day alike in the Army, with its " serf -like rank- 

 and-file " and its "arrogant, overbearing caste of 

 officers," and in "an almost serf -like people," 

 which bows to the despotism of the Kaiser, the 

 Junker, and the lords of commerce and finance 

 "as to the gracious rule of a divinelv instituted 

 * State.' " 



Having been Prussianised themselves, the 

 Germans have sought solace in the dream of 

 "forcibly Prussianising a greater Germany which 

 would expand at last into a world-wide empire." 

 They have also sought to make their dream come 

 true. "The aggressive egoism of an over-docile 

 people is the torch which has set the world ablaze. " 

 The blaze has given the world a glimpse of the 

 pernicious way in which over-docility may deaden 

 and brutalise a people. We wish it had left them 

 less effective ! That it will eventually betray 

 Germany in the field is evidently the author's ex- 

 pectation, which we cannot but share. We wish 

 again, however, that the symptoms of material 

 Nemesis were a little more convincing than they 

 are as yet, for the temporan,- success of the 

 thoroughness of the ultra-docility which Mr. 

 Holmes so vigorously damns remains as the active 

 cause of incalculable wastage and misery, and as 

 a terribly disquieting menace to civilisation. 



While we are inclined to regard the author's 

 survey as one-sided, and his interpretation in 

 terms of the " ultra-docility " formula over- 

 strained, we feel that he has powerfully pre- 

 sented part of the truth, and driven honie the 

 salutary moral : Fas est et ah haste doceri. 



(2) This is a beautifully printed lecture on 

 "The War and Medical Thought," delivered in 

 December, 1914, as a presidential address to the 



