April 'i^, 1890] 



NA TURE 



509 



constant coefficients. After the appearance of Fuchs's 

 second memoir, many mathematicians, particularly in 

 France and Germany, including Fuchs himself, took up 

 the subject, which, though still in its infancy, now pos- 

 sesses a very large literature." 



As happens in such cases, these memoirs have to be 

 dug out of journals and publications of learned Societies 

 before the student can be put in possession of results 

 obtained. It is for this labour of research, and then for 

 the arrangement in due sequence of theorems, that the 

 reader has to thank Dr. Craig.^ Even in the first two 

 chapters, where most of the results are old, the treatment 

 is comparatively new, being founded upon papers by 

 Laguerre {Comptes reiidus, 1879), and upon memoirs, or 

 works, by Briot and Bouquet and Jordan ; reference is 

 also made, in connection with a proof by Jordan, to a 

 paper by Picard (^Bulletin des Sciences Math., 1888). 

 Here we may note that the author reserves an account of 

 the investigations of Laguerre, Halphen, and others, from 

 a still higher point of view, to a subsequent volume. 



This first instalment discusses principally Fuchs's type 

 of equations, but accounts are given of the researches of 

 Frobenius (chapters iv., viii.), Markoff, Heun, Riemann, 

 and Humbert (chapter vi.), Thome (chapter ix.), Halphen 

 (chapter xii.), Forsyth's canonical form and associate 

 equations, Brioschi, Lagrange's adjoint equation, Hal- 

 phen's adjoint quantics and Appell's theorem (chapter 

 xiii.), and Picard (chapter xiv.). An account, due to 

 Jordan, is given of the application of the theory of sub- 

 stitutions to linear differential equations (chapter iii.). 

 Many points are touched lightly here, a fuller develop- 

 ment being held in reserve. A prominent feature is the 

 reproduction (chapter vii ) of a thesis by M. E. Goursat 

 on equations of the second order satisfied by the hyper- 

 geometric series. This consists of two parts. The first 

 part gives an application of Cauchy's theorem, and rela- 

 tions between Kummer's (24) integrals, an application to 

 the complete elliptic integral of the first kind, and 

 Schwarz's results. The second part discusses the trans- 

 formations of the hypergeometric series. Tannery's 

 theorem, and some other points, the article closing with a 

 collection of 137 transformations due (apparently) to 

 Kummer. 



The pages bristle with references to original sources, 

 so that, as we have already indicated, this treatise is an 

 invaluable handy-book to what has been done in this 

 field. 



One more word : there is no collection of examples for 

 solution on the Cambridge model, but the work is strictly 

 on the lines of a French or German treatise. 



The book itself is very elegantly turned out. 



THE BACTERIA OF ASIATIC CHOLERA. 

 The Bacteria of Asiatic Cholera. By E. Klein, M.D 

 (London : Macmillan and Co., 1889.) 



SO masterly and complete was the account which 

 Koch gave in 1884 of the comma-bacillus, which 

 he held to be the virus of cholera, that but little, if any- 

 thing, has been added to our knowledge of its mode of 



' For instance, he obtains certain forms in the same way that Fuchs 

 •obtained them, '' if for no other reason than that of the desirability of. 

 ■developing the subject in historical order " (p. 64). 



growth, of its reaction to dyes, or of its life-history. As 

 might be expected, the assiduity of many observers, now 

 it has been directed to the subject, has led to the dis- 

 covery of many other bacilli, whieh may be described as 

 comma- shaped. But, so far, no bacteriologist, who has 

 had his observations corroborated by other observers, 

 has proved that any of them are indistinguishable in all 

 their physical characters, whether in appearance, in re- 

 action to dyes, or in their mode of growth, &c., from the 

 ^choleraic bacillus. So far as is known, animals are i ot 

 susceptible to cholera. If Asiatic cholera could be in- 

 duced by inoculating with pure cultivations of choleraic 

 comma-bacilli, then beyond a doubt they would be the 

 nera causa, or, in other words, the contagium of cholera ; 

 but this step in Koch's argument was wanting, probably 

 for the above-named reason, and is likely to remain so : 

 the experimental inoculations of guinea-pigs which have 

 taken place being by no means conclusive. 



The present volume is a valuable and most trenchant 

 criticism of every step of Koch's argument, and may be 

 said to contain everything that can at present be said 

 against Koch's theory, of which the author is the most 

 active opponent. 



The author commences with an account of the various 

 comma-shaped bacilli which are at present known, and 

 there are well- recognized characteristics which distinguish 

 them from the first form, in all of them, except in those 

 which depend upon solitary observations. 



The following is the list of comma- shaped bacilli with 

 the names of their discoverers : — 



(i) Koch, in Asiatic cholera ; ^ to | the length of 

 tubercle bacilli, but thicker and curved. (2) Finkler and 

 Prior, in cholera nostras ; but Koch and Frank failed to 

 demonstrate these in typical cases. They are thicker 

 and longer than (i). In 10 per cent, gelatine, the growth 

 is broad and conical, liquefying the gelatine more rapidly- 

 (3) Lewis, in the fluid of the mouth, thicker than (1) 

 Klein only twice has succeeded in growing them ; every 

 one else has failed. (4) Miller, in some cases of caries 

 of the teeth, similar to (2). (5) Kuisl, in human faeces 

 similar to (2). (6) Deneke, in stale cheeses. The growth 

 on gelatine is similar, but they will not grow on potatoes. 

 (7) Klein, in some cases of diarrhoea, especially in mon- 

 keys. They grow differetly in gelatine, and cause it to 

 smell offensively. (8) Ermengen and others, in the in- 

 testines of guinea-pigs, pigs, rabbits, horses, &c., but they 

 will not grow in 10 per cent, gelatine. (9) Lingard, two 

 kinds in a case of noma, the smaller of which is said to 

 have been very similar to the choleraic one. (10) Weibel, 

 various forms in mucus, but their mode of growth is 

 distinct, (il) Gamaleia, in a fatal fowl disease, which 

 was prevalent at Odessa. He did not distinguish them 

 from (l).- (12) Klein, in the intestines of a monkey with 

 diarrhoea. The organisms were smaller, but the growth 

 was similar to (i). 



Klein lays great stress upon the difficulty there is in 

 demonstrating the presence of the bacilli in the walls of 

 the intestine in cases of cholera, and thinks that they 

 are not present in the parts which are still alive, but only 

 where the tissue has died ; moreover they are absent 

 from the blood. 



The bacilli are most readily found in the mucous 

 flakes ; and in the presence of faecal matter they are 



