December 15, 1910] 



NATURE 



197 



convinced him that the colour-changes which these 

 reptiles undergo with such rapidity are not, as often 

 believed, in harmony with their surroundings, but are 

 regulated chiefly by light, temperature, excitement, 

 fright, or health. \Ve here reproduce a partial list of 

 these experiments on the common chameleon : — 



Specimen A. Placed in the sunlight so that but one 

 side of the lizard was exposed to the rays. 



Specimen B. Placed in the sunlight at an angle to 

 entirely suffuse the reptile with the rays. 



Specimen C. Placed in a dark box ; temperature, 

 73° F._ 



Specimen D. Placed in a dark box ; temperature, 

 50° F. 



After fifteen minutes, the following results were 

 noted :— 



Specimen A. Was a dark brown on the side that had 

 been exposed to the sun ; the shadowed side was a pale 

 brown, mottled with green. 



Specimen B. A uniform brown, deeper than the 

 dark side of specimen A. 



Specimen C. When the cover of the box was drawn 

 the lizard emerged in a brilliant coat of green. 



Specimen D. Crawled sluggishjv from the cold 

 vjuarters. Its colour was a uniform slaty-grey. 



One curious effect of sunlight and shadow was 

 noticed. A specimen had been basking under a 

 coarse wire grating. Becoming frightened at the 

 approach of the observer, it changed its position. On 

 the dark brown body was what had been the shadow 

 of the grating, brilliantly imprinted in pale yellow. 

 Within half a minute this pattern had entirely faded. 



The book is copiously illustrated with reproductions 

 of photographs taken by the author from living 

 specimens, and most of them are of high excellence. 

 In some cases, however, the reduction is too great, 

 such figures as those of the European lizards and 

 the glass-snakes and slow-worm (plates xxxia and 

 xxxvii.) being, from this cause, practically useless. 

 The snake figured on plate Ixxvii as Cerastes vipera, 

 and stated to measure about two and a half feet, is a 

 hornless Cerastes cornutus. The author appears to be 

 unaware of the existence of such hornless specimens, 

 otherwise he would not have written (p. 328) that it is 

 "impossible to mistake the horned viper," and that 

 C. vipera is, but for the absence of horns, much like 

 C cornutus. A three-colour process figure of the 

 rhinoceros viper, "the most beautifully coloured of all 

 poisonous snakes," is given as a frontispiece. 



G. A. B. 



THE CALCULUS OF VARLiTIONS. 

 Lccons sur le Calcul des Variations. By Prof. J. 

 Hadamard. Tome premier. Pp. viii + 520. (Paris: 

 A. Hermann et Fils, 19 10.) Price 18 francs. 

 ^VrO one could be more competent than M. Hada- 

 ^ ^ mard to deal with the calculus of variations, and 

 when this work is completed it will be a most valuable 

 exposition of the present state of the subject. It is 

 significant that in the first hnes of his preface the 

 author expresses the view that the calculus of varia- 

 tions is only a first chapter of the functional calculus 

 icalcul fonctionnel) of Volterra, Pincherle, &c., and 

 he gives, in fact, a short chapter on this new theory 

 (pp. 281-312). But the analysis, in this volume, is 

 mostly of a more familiar kind. 

 NO. 2146, VOL. 85] 



In fact, the first step in any actual case that natur- 

 ally presents itself is still the classical one of 

 Lagrange, by which we obtain a differential equation, 

 or a set of differential equations. For simplicity, sup- 

 pose the varied integral to be 1 f(x, y, y')dx, then the 



differential equation is of the second order, and its 

 I solution is said to form a family of extremals. Sup- 

 posing that the limiting values of x and the corre- 

 sponding values of y to be given, then in the general 

 case we may expect to find one extremal satisfying 

 the terminal conditions. But it by no means follows 

 that this curve really makes the given integral a 

 maximum or minimum ; an example due to Scheeffer 

 is given on p. 45, which brings out the point very 

 clearly. In this case the extremal found from the 

 differential equation is y = o, and the corresponding 

 value of the integral is o; nevertheless, analytical 

 curves can be drawn, as close as we please to y = o, 

 which make the integral negative. 



In any case, a solution obtained from an extremal 

 is only a relative one ; that is, the extremal gives a: 

 maximum or minimum value r>f the integral relatively 

 to adjacent paths. And here it is important to define- 

 what we mean by adjacent, a fact first fully realised ■ 

 by Weierstrass, whose definition of adjacency of the 

 pth. order is given on p. 49. W^e may have, for in- 

 stance, two curves each passing through the terminal 

 points A, B, and as close together as we please, but 

 one may be of continuous, the other of discontinuous 

 curvature. Now, if we have a varied integral involv- 

 ing higher differential coefficients than y', we must 

 exclude curves of discontinuous curvature, otherwise 

 the problem becomes meaningless, and similarly in 

 other cases. 



After the limitations of the problem have thus been 

 touched upon, book ii. deals with the first variation,, 

 and the conditions of the first order, including variable 

 limits. Among other interesting points we have 

 Weierstrass 's transformation to homogeneous co- 

 ordinates, a discussion of foci (points on the envelope 

 of a family of extremals), and two very useful inno- 

 vations due to M. Hadamard. If / f{x, y, y')dx is the 

 varied integral, the figurative is defined to be the 

 curve f{x, y,y') — u, in which u,y' are regarded as 

 current coordinates, and x, y as constants. The 

 figuratrix is defined as the polar reciprocal of the 

 figurative with respect to x' + y' = i. By means of 

 these curves the author is able to put various analy- 

 tical conditions into a vivid geometrical shape. It 

 mav be added that book ii. contains the discussion of 

 various classical problems, such as brachistochrones, 

 least action, the Hamiltonian equations of dynamics, 

 &c. 



Book iii. introduces the second variation, and goes 

 more deeply into the methods of Weierstrass, as well 

 as those of Jacobi, Clebsch, Hilbert, Kneser, and 

 others. We arrive ultimately at a statement, in 

 various forms, of sufficient conditions for a minimum 

 (pp. 389, 397), deduced mainly from the properties of 

 a pencil of extremals, and a brief discussion of the 

 necessary conditions, illustrated by examples (chapter 

 iii.). The remaining- chapters deal with variable 

 limits, discontinuous solutions, Osgood's theorem in 



