October lo, 1895] 



NATURE 



567 



p. 38 uc read, " The influence of sea air in causing 

 ana'mia is apparent on many parts of the coast," and on 

 p. 47, " It may be stated that the infrequency of anxmia 

 in the local inhabitants is no doubt due to their proximity 

 to the Atlantic." 



To instance the difficulty, which frequently presents 

 itself, of arriving at just conclusions from the statistical 

 information acquired by the committee, let us ask our- 

 selves what inference may justly be drawn when the 

 phthisis rate is high in certain health resorts. It is very 

 properly pointed out that much of this excess is doubtless 

 due to phthisical immigrants to a spot which is known 

 to be congenial to phthisical patients. Quite true I But 

 if we cannot ascertain to 7vhal extent \\\<i rate is influenced 

 by phthisical immigration, how is one to know whether 

 the local conditions, per se, are favourable or not to the 

 disease in question ? It is conceivable, in this relation, 

 that certain limited areas of England with comparatively 

 mild and equitable cliniatcs have now a native population 

 strongly predisposed to phthisis, from the fact that their 

 ancestors were originally phthisical immigrants attracted 

 to the spot ; so that even if it were practicable that the 

 vital statistics of visitors could be separately compiled, 

 the local and climatic advantages or disadvantages of 

 the area in respect of this disease could never be put 

 upon a scientific basis from vital returns alone. It is 

 well known, moreover, that deductions drawn from 

 meteorological data on the score of the suitability of 

 the various areas for the residence 6f those suffering 

 from different diseases, must be made with many reserva- 

 tions, that the subject does not admit of generalisations ; 

 for, !/!ter iilia, the suitability of the climates of certain 

 health resorts for different patients is governed to such 

 an extent b\- that wonderful personal factor that makes 

 the same spot bracing to one and relaxing to another, 

 benevolent to a certain disease in one and malignant to 

 that disease in another, that frequently the individual 

 can only arrive at the conclusion as to which area suits 

 him best by an actual personal experiment. And thus 

 it comes about that perhaps, after all, the surest lines 

 upon which a physician can act, are in the main em- 

 pirical as to his patient. We have lived long enough 

 in these islands to know by experience which are the 

 •warmest, driest, and most sheltered spots, which are the 

 ■dampest, and which aie the most bracing and relaxing, 

 and it is quite a question whether meteorological data 

 Avill help the physician much farther. He will generally 

 select for his patient what has been proved by the ex- 

 perience of many generations to be a congenial site, and 

 nothing short of a cautious experiment with the patient 

 himself will suffice to tell him which of several alternative 

 sites suits his patient best ; but to this end the experiences 

 and views of other practicising physicians would be of 

 immense value, and one is templed to ask whether a 

 work embodying and summarising as many as possible 

 of these experiences would not serve e\ en a more useful 

 purpose than the first 500 pages of this book. 



The chapters dealing with the medicinal waters of (/reat 

 Britain are well written, useful, concise and impartial. 



The committee hopes to deal in a further report with 

 the climatology of the remaining districts, and with those 

 mineral springs which are not included in the present 

 'lolume. 



NO. T354. VOL 52] 



OUR BOOK SHELF. 



Par 



Abrci^i' tie la Theorie lies Fonctions Elliptiques. 

 Charles Henry. 124 pp. (Paris : Nony, 1895.) 



An introductory course of elliptic functions, intended for 

 those who have a fair acquaintance with integral calculus, 

 should consist of three stages. In the first stage the 

 subject would be approached as a development of integral 

 calculus, the addition theorem and periodicity obtained, 

 and a large number of applications made to problems 

 whose solutions can be expressed in the notation of 

 elliptic functions. Difficulties of the multiple interpreta- 

 tion of the square roots of variable functions would be 

 pointed out, and left. In the second stage an elementary 

 introduction to the modern descriptive theory of functions 

 of a complex variable would be furnished, containing a 

 fairly full account of the theory of doubly periodic 

 functions, illustrated at every stage by examples from the 

 functions whose existence has been foreshadowed in the 

 first stage. The third stage would be a systematic de- 

 velopment of the elliptic functions, with the help of the 

 elementary theory of functions, finishing, not beginning, 

 with the differential etiuation and the applications to in- 

 tegral calculus. Such a course would require at least 

 twenty-fi\e hour-lectures, and the unfamiliar character of 

 the second and third stages would make a careful revision 

 necessary. 



The present little volume is concerned with the third 

 stage ; on the whole, there can be no doubt that it is the 

 most suitable handbook which has yet appeared for the 

 use of teachers engaged in such a course as sketched 

 above. The eUiptic functions are obtained by the infinite 

 double series iox p{it) ; and certainly the idea is the right 

 one, though it is easier to begin with the series for /'(//). 

 The differential equation is hence obtained, and the 

 foUowing chapter attempts to establish the functions 

 on that basis. It seems preferable that this should be 

 postponed, and treated only by Riemann's methods. 

 Chapters iii. and iv. introduce the functions f k and <ru, 

 as is quite proper ; but it would seem much better that 

 the addition equation, obtained in chapter v., should be 

 obtained independently of the o- functions, and by Abel's 

 method, with the help of a plane cubic curve. The 

 functions c7-,(//), <tJii), (rA"), are then obtained, and hence 

 it is proved that the functions s!pu-ey, . . . arc single- 

 valued functions of //. It is a distinct step in the right 

 direction to make the statement that these functions 

 ■Jpii-ei,. . . are single-valued; but the fact ought to 

 be obtained before, and independently of, the investi- 

 gation of their actual values. The same remark holds in 

 regard to the functions en k, dn u ; if x—sn u, it ought 

 to be shown that J i —x- is single-valued before its 

 actual value is obtained, and the r emark emphasised by 

 proving that such a function as ^/(i —sni/)(i -i-sni/) is 

 equally a sing^le-\alued function of //. The fact, which 

 is obtained, that all doubly periodic functions are 

 rationally expressible by / u and /' ;/, ought to be com- 

 pared with the fact that all doubly periodic functions are 

 rationallv expressible by s/i it and en it tin 11 ; and it ought 

 to be clearly seen that when we are dealing with Jacobi's 

 functions, en u is no more a function of the same kind as 

 sn u than is .Jpu - c, of the same kind as / it when we 

 are dealing with Weicrstrass's functions. In these two 

 cases respectively, (v; // and -Jpu-e^ arc factoriiii {\mc- 

 tions, which ought to be carefully distinguished from 

 the two fundamental functions whereby the algebraical 

 irrationality under consideration is resolved. 



With these criticisms, and the remark that the accounts 

 of the transformation and of Jacobi's 6 functions are not 

 so full as one desires, we may conclude, strongly recom- 

 mending all who desire a useful class book, to which, 

 however, many explanations and illustrative examples 

 must be supplied, to adopt the book. H. F. B.\Ki:R. 



