266 



NATURE 



[July 21, 



Miiller, and ending with those now generally in use, is 

 excellent ; although it is evident from what is here laid 

 down, that our classification of micro-organisms is as 

 yet to a large extent empirical, and that there is great 

 need for a classification constructed on a thoroughly sound 

 and scientific basis. The principles of sterilisation and 

 pure cultivation are given succinctly but very clearly. 

 The section devoted to the heat-resisting bacteria, their 

 place in nature, and their importance in the fermentation 

 and food-stuflf industries, is one of considerable interest. 

 The principal organisms in this group are described as 

 the Bacillus subtilis and its congeners, the Clostridium 

 butyricum, the genus Granulobacter, and various other 

 organisms associated with the butyric acid fermentation, 

 the fermentation of cellulose, the "retting" of flax and 

 hemp, and the production of rancidity of fats. 



The relation of the study of the life-history of these 

 various organisms to the preservation of milk, meat, 

 eggs, vegetables and fruit is fairly carefully considered, 

 as are also the lactic fermentation and the allied decom- 

 positions, special stress being laid on the production of 

 optically active organic compounds by fermentation, on 

 the artificial souring of cream, the coagulation of milk, 

 and on the importance of the part that various lactic 

 acid bacteria play in the processes of distilling, brewing 

 and vinification ; and in the preparation of fodder, the 

 making of brown hay into sweet ensilage and sour fodder. 

 Then the work done by bacteria in tanning, in the manu- 

 facture of sugar in the conditions known as "ropiness" 

 in milk, wine, beer, and other liquids are all somewhat 

 fully and interestingly treated. A special section is de- 

 voted to the decomposition and transformation of organic 

 nitrogenous compounds ; this, of course, constitutes a 

 very important part of the work, and, in conjunction 

 with the section on oxidising fermentation, affords a very 

 large amount of information on the bacterial processes 

 involved in the breaking up of various organic com- 

 pounds. It is interesting to note how closely these 

 processes are associated with those of fermentation of 

 cheese and of similar proteid substances. 



Altogether this volume, the first of two, is an exceed- 

 ingly interesting and valuable contribution to the study 

 of technical mycology. The work of translation is well 

 done, but there are one or two slips which might with 

 advantage be corrected in future editions : for instance, 

 " typhus " is throughout used for " typhoid," this, of 

 course, being a literal translation of the German typhus 

 without the term abdominalis, which is always added to 

 indicate our typhoid fever. It need scarcely be mentioned 

 that the work will probably be hailed by English workers 

 with gratitude, but we may point out that the term 

 "mycology" will convey to the general reader very little 

 idea as to the scope of the work. Many years ago a 

 work was published in this country to which the title 

 "Pathological Mycology" was given, a work which was 

 largely overlooked because of its title. Since then this 

 same title has been used abroad, where the significance 

 of the word appears to be more fully appreciated. We 

 think the translator would have been wise had he selected 

 some title more generally " understanded of the people " 

 for what, after all, must to a certain extent be a popular 

 work. There will, however, be a considerable demand 

 for this book amongst those who are engaged in patho- 

 NO. 1499, VOL. 58] 



logical and technical bacteriology, who, of course, will 

 appreciate both the title and the work ; but the translator 

 must expect to find that some, at least, of his possible 

 readers will pass over this book simply because they do 

 not understand the title. 



Messrs. Griffin have done their part in a thoroughly 

 workmanlike fashion, and we congratulate both author 

 and translator in having their work placed so well before 

 the reading public. 



PARTIAL DIFFERENTIAL EQUATIONS. 

 Legons sur V integration des equations aux derivdes 

 partielles du second ordre d, deux variables indipend- 

 antes. Par E. Goursat. T. I. pp. viii-f226; T. II. 

 pp. 344. (Paris : A. Hermann, 1897, 1898.) 



ADIFFERENTIALequation, in its usual form, states 

 an analytical problem with a certain assumption 

 as to the form of the answer. It implies the existence of 

 a dependent variable, capable of being differentiated so 

 far as the order of the equation indicates ; and the 

 solution of the equation consists in discovering a relation 

 among the variables, free from differential coefficients, 

 such that the given differential equation may be derived 

 from it. The question at once arises : what is the most 

 comprehensive form of solution? Is it possible in every 

 case to define an integral relation connecting the variables 

 equivalent to the differential equation in the sense that 

 not only is the differential equation derivable from it, but 

 every possible relation consistent therewith is included 

 as a particular case in the integral equation ? In the 

 early days of the infinitesimal calculus it was observed 

 that ordinary differential equations could be obtained by 

 eliminating constants ; while partial differential equations 

 could be derived by the elimination of constants or of 

 arbitrary functions. In some cases the reverse process of 

 starting with the differential equation and arriving at an 

 integral relation, involving arbitrary constants or func- 

 tions, or both, was found to be practicable ; and it came 

 to be taken for granted that integral relations of this 

 kind always existed, the only difficulty being that of 

 discovering them. 



But, with the advance of function-theory, the peculiar 

 difficulties of the subject have gradually become more 

 evident. It is true that, with regard to ordinary differ- 

 ential equations and partial differential equations of the 

 first order, the general form of solution has been estab- 

 lished, and the hypothesis of the earlier mathematicians 

 justified ; but when we come to partial differential equa- 

 tions of the second and higher orders, the aspect of the 

 problem is radically changed. In most cases it is hope- 

 less to attempt to assign an explicit form of the general 

 integral, or even to prove its existence ; and we have to 

 content ourselves with the study of solutions subject to 

 certain special limitations. Thus we have the problem 

 of Dirichlet in the theory of potential ; or again the 

 problem of Cauchy, which forms the leading idea of 

 M. Goursat's original and fascinating treatise. 



To explain what this means, let us take the case of an 

 equation of the second order with two independent vari- 

 ables, say ^(.r, y, Siptq, f, s, t) = o, the notation being as 

 usual. Assume x,y, z,p, q functions of a single variable,, 

 subject solely to the condition dz — pdx -h qdy ; we thus 



