April 6, i 899] 



NA TURE 



533 



La Photographie Aniinec. By Eug. Tiutat, Director of 



the Natural History Museum, Toulouse. Pp. xii + 185. 



(Paris: Gauthier-Villars, 1S99.) 

 This volume, introduced by a preface by M. Marey, the 

 well-known chronophotographer of animals and human 

 beings in motion, for purposes of study, will be found 

 useful to all interested in the subject of animated photo- 

 graphy. 



The author devotes the opening chapter to a short 

 review of the history of the subject, explaining the 

 application of the phenomenon of persistence of vision in 

 such early instruments as the phenakisticope and zootrope 

 of Plateau and Clerk Maxwell. 



He then traces the evolution of the apparatus from the 

 multiple cameras of Muybridge, Anschutz, Londe and his 

 own to the first employment of a fixed plate by M. Marey, 

 and then to the continuous band machines of Marey, 

 Edison, Demeny and others. In this chapter will be 

 found well-illustrated descriptions of most of the French 

 machines which have proved successful. 



The third and concluding chapter deals with the 

 various manipulations necessary for obtaining the photo- 

 graphs, and afterwards exhibiting them. The operations 

 of exposure, development, and printing of the positive 

 film are lucidly explained, and then details are given for 

 the management of the film in the lantern. 



There is no doubt of the usefulness of the treatise, but 

 its value is somewhat lessened by the descriptions being 

 almost entirely confined to French apparatus, the author 

 giving no signs of being familiar with the successful 

 machines which have been produced outside his own 

 country. 



LETTERS TO THE EDITOR. 



\The Etiitor docs not hold himself responsible for opinions ex- 

 pressed by his correspondents Neither can he undertake 

 to return., or to correspond with the writers of, rejected 

 manuscripts intended for this or any other pai-t of NATURE. 

 No notice is taken of anonymous communications. ] 



The Interferometer. 



The questions raised by Mr. Preston (Nature, March 23) 

 can only be fully answered by Prof. Michelson himself; 

 but as one of the few who have used the interferometer in ob- 

 servations involving high interference, I should like to make a 

 remark or two. My opportunity was due to the kindness of 

 Prof Michelson, who some years ago left in my hands a small 

 instrument of his model. 



I do not understand in what way the working is supposed to 

 be prejudiced by " diffraction." My experience certainly sug- 

 gested nothing of the sort, and I do not see why it is to be 

 expected upon theoretical grounds. 



The estimation of the " visibility " of the bands, and the de- 

 duction of the structure of the spectrum line from the visibility 

 curve, are no doubt rather delicate matters. I have remarked 

 upon a former occasion {Phil. Mag., November 1892) that, 

 strictly speaking, the structure cannot be deduced from the 

 visibility curve without an auxiliary assumption. But in the 

 application to radiation in a magnetic field the assumption of 

 symmetry would appear to be justified. 



My observations were made with a modification of the original 

 apparatus, which it may be worth while briefly to describe. In 

 order to increase the retardation it is necessary to move back- 

 wards, parallel to itself, one of the perpendicularly reflecting 

 mirrors. Unless the ways upon which the sliding piece travels 

 are extremely true, this involves a troublesome readjustment 

 of the mirror after each change of distance. The difficulty is 

 avoided by the use of a fluid surface as reflector, which after 

 each movement automatically sets itself rigorously horizontal. 

 If mercury be contained in a glass dish, the depth must be con- 

 siderable, and then the surface is inconveniently mobile. A 

 better plan is to use a thin layer standing on a piece of copper 

 plate carefully amalgamated. A screw movement for raising 

 and lowering the mercury reflector is still desirable, though not 

 absolutely necessary. Ravleigh. 



NO. 1536, VOL. 59] 



Theory of Functions. 



In his review of our book on " .\nalytic Functions'' 

 (Nature, February 23), Prof. Burnside makes three specific 

 charges of inaccuracy ; we shall show that the inaccuracy 

 is his, not ours. 



(i) One charge relates to the difference of two convergent 

 series. There is an elementary and well-known theorem which 



states that the difference of two convergent series 2a„ and 2/',, 



1 1 



is equal to 2[iZ„ - b„), no matter whether the convergence of the 



series be unconditional or conditional. Prof Burnside has, 

 then, fallen into a very serious error when he says of this very 

 operation of subtraction that " the rearrangement involved is 

 one which cannot be used with conditionally convergent series, 

 as indeed the authors have shown most clearly in an earlier 

 chapter." We must add that there is no "rearrangement," 

 and that we have tried in § 68 to put the reader on his guard 

 against this very error of Prof Burnside. 



(2) A second charge relates to infinite products. In § 109 we 

 consider a certain infinite product n(i - a„) ; in regard to this 

 product. Prof. Burnside complains that we have not explained 

 " what is implied in calling such a product convergent." As a 

 matter of fact we treat an infinite product as an instance of an 

 infinite sequence, and convergence for infinite sequences has 

 been already explained in § 47. He falls into another inac- 

 curacy when he says that "if 2o,i is greater than unity, all that has 

 been proved is that n(l - o„) is less than unity and greater than 

 some definitive negative quantity." We have proved much more 



than this, namely that there is a limit for the numbers n( I - a„), 



1 

 when K tends to infinity (see § 45). 



We did not intend to go into the case where the sequence 

 associated with an infinite product converges to zero, because 

 there is as yet no final agreement as to whether the product is 

 or is not to be called convergent in this case. The product in 

 § log does not converge to zero. Prof Burnside does not 

 allude to this point ; but we should like, nevertheless, to take this 

 opportunity of saying that we ought to have added a proof that 

 the convergence of 2a„ excludes this special case, instead of 

 assuming that the reader knows the proof, as given, for instance, 

 in Hobson's "Trigonometry." 



(3) The third charge relates to our use of the word "infinity " 

 on p. 3. This word "infinity," in the earlier parts of the 

 higher arithemetic, has but one accepted meaning ; to quote 

 the words of M. Tannery, "la notion de I'infini dont il ne 

 faut pas faire mystere en mithematiques se reduit a ceci ; apres 

 chaque nombre entier il y en a un autre." We have used the 

 word "infinity" in this, its legitimate sense. Failure to per- 

 ceive the " variable " character of infinity has led to many mis- 

 conceptions in the past. We cannot understand Prof Burnside's 

 objection except on the supposition that he has, for the moment, 

 confused this "variable" infinity with the discredited "con- 

 stant " infinity. 



On the score of accuracy we wish to point out that we gave 

 two chapters to elliptic functions, not three, as the reviewer 

 states ; and that Log x is not define I (the italics are the re- 

 viewer's) by means of a piece of string and a cone. We define 

 the logarithm by means of an equiangular spiral, in a way some- 

 what similar to that used in Cliff'ord's "Common Sense of the 

 Exact Sciences," and we indicate, incidentally, a mechanical 

 construction of the curve. 



It is always an ungracious task to reply to a review, especially 

 when it is in general appreciative, and written by a mathematician 

 of acknowledged standing ; but in the circumstances we fell that 

 we had no alternative. We believe that Prof Burnside will be 

 the first to recognise that his specific criticisms are based on 

 misconceptions. J. Harkness. 



Philadelphia, March 14. F. Morlev. 



The criticism on the pissage quoted from p. 3 of the book by 

 Profs. Harkness and Morley (Nature, February 23, p. 347) 

 turns on the fact that, in dealing with number divorced from 

 measurement, the authors have used the phrase " an infinity of 

 objects" without an explicit statement of its meaning. I am 

 not sure that I understand the pissage in their letter which refers 

 to this point ; but it seems to me to imply that the distinction 

 between "finite" and "infinite " is one which does not require 

 definition. This is not the only accepted view. It is not, for 



