252 



NATURE 



{Jan. 1 8, 1877 



gas series ; the nitrogen being evolved from the albu- 

 minous matter of the wort. By diminishing the pres- 

 sure the amount of evolved hydrogen increased, and 

 with this an increase in the amount of acetic acid and 

 aldehyde. These products, though very small compared 

 with the alcohol and carbonic acid, the chief resultants of 

 ferment action are yet sufficient to account for the 

 etherial odour of a fermenting tun. Pasteur had pre- 

 viously noticed the production of minute quantities of 

 volatile acids. On electrolysing a weak aqueous solution 

 of invert sugar, Brown obtained carbonic acid, hydrogen, 

 and oxygen, and at the same time an appreciable quan- 

 tity of aldehyde and acetic acid, together with a small 

 quantity of formic acid. It may be that water is decom- 

 posed in fermentation in small quantities precisely as 

 occurs in ordinary vegetation ; though highly probable, 

 we have, however, no definite facts in support of the 

 assumption. Our present knowledge of the chemistry of 

 fermentation is somewhat vague and general, and much 

 remains to be done before we shall be enabled by 

 purely physical means to decompose sugar so as to 

 produce the results brought about by the yeast cell. 



As we progress in knowledge, so does our power to 

 imitate the products of life-action increase, and assuredly 

 the time will arrive when alcohol will be produced by 

 simple physical or chemical means. For many a year to 

 come, however, we must continue to depend upon the 

 wonderful organisms known as yeast. Their life history 

 and action on liquids have been elucidated by the genius 

 and patient toil of Pasteur, and he has enabled us to 

 select such ferments as are required to produce the result 

 desired, and hence we are no longer the sport of chance. 



The brewer and wine-maker are not alone in the debt 

 due to the illustrious Frenchman whose work we have 

 briefly examined ; we are all interested in the far wider 

 field of germ action opened out by him, whereby many 

 a disease of man hitherto as dark and unexplained 

 as was fermentation, will be brought under the illumin- 

 ating light of his teaching. 



He who has thus shown us the key whereby we may 

 open the locked-up secrets of nature may rest assured of 

 the gratitude of his fellow-men, given with the greater 

 earnestness and respectful sympathy from the knowledge 

 that our guide has impaired health and sight in his labours 

 for others. 



France has given many a great name to the roll of 

 fame, but none more noble or more worthily inscribed 

 thereon than that of Pasteur. Charles Graham 



CAYLEY'S ELLIPTIC FUNCTIONS 



Elementary Treatise on Elliptic Functions. By Arthur 

 Cay ley. (Cambridge : Deighton, Bell, and Co.). 



THIS is a book thoroughly worthy of the great name 

 of its author. It is difficult to know which to 

 admire most, the grasp of the subject, the extreme sim- 

 plicity of its exposition, or the neatness of its notation. 

 It will, we think, at once take its proper place as the 

 leading text-book on the subject. 



In regard to notation, it seems to us to be thoroughly 

 good throughout, not only in respect of the adoption of 

 Gudermann's suggestion of the very short forms sn, en, 

 dn, for the sine, cosine, and elliptic radical of the ampli- 



tude of the function of the first kind, but throughout. 

 In particular, we note an important typographical sim- 

 plification in the suppression of the common denominator 

 in long series of fractional formulas, the denominator 

 being given once for all, and its existence in each separate 

 formula merely indicated by the sign of division {—) 

 This is a simplification, some equivalent of which we 

 ourselves, and probably most of those who have worked 

 at elliptic functions, have used in our private papers ; but 

 it is a new thing, and a very good thing, to see it intro- 

 duced in a systematic form in a printed book. 



Another very useful feature belonging to mechanical 

 arrangement is, that the first chapter contains a general 

 outline of the whole theory, so that its perusal enables the 

 reader to see at a glance the plan and intention of the 

 work. He is thus enabled at once to bring intelligent at- 

 tention to bear upon his reading, instead of being dis- 

 tracted by the wonder as to what it is all driving at. 



The intention of the work is, firstly, the direct dis- 

 cussion and comparison of the three forms of elliptic 

 integral, and of the doubly periodic functions which are 

 regarded as the direct quantities of which these integrals 

 are inverse functions. Then the auxiliary functions 

 Z, H, and H are taken up in an elementary form, and 

 after this the transformation of the elliptic functions, 

 by division or multiplication of the primary integral, with 

 the corresponding change of modulus and amplitude. 

 These are very fully and clearly discussed. In particular, 

 the connection between the transformation of the radical 

 in the elliptic integrals, and the formula; of multiplication, 

 is clearly brought out. Legendre had left this as a very 

 puzzling, although necessary, inference, which he scarcely 

 stopped to discuss. After this comes a discussion of the 

 q functions, with a further discussion of the functions H 

 and H, and then some miscellaneous developments. 



The work is strictly confined to elliptic functions and 

 their auxiliaries. The more general theories of Abel and 

 Boole find no place in it, nor is there any general dis- 

 cussion of single and double periodicity such as forms the 

 foundation of the work of Messrs, Briot and Bouquet. 

 There are but few examples of the computation of par- 

 ticular values of the elliptic functions, and no account of 

 general methods of computation, either of isolated values, 

 or of tables, or of the arithmetic connected with ihem ; 

 nor are ultra-elliptic functions touched upon. The geo- 

 metrical applications or illustrations of the elliptic inte- 

 grals and functions are but meagre, and no mechanical 

 applications are given. 



The arithmetical work is quite rightly omitted. That 

 will find a much better place in the hand-book or intro- 

 duction which will doubtless accompany or follow the 

 great tables of elliptic functions now being printed for 

 the British Association. There is, however, one point 

 which we think it an omission to notice, and that is 

 the solution of the addition equation by means of auxi- 

 liary angles. (See Legendre, "Traitd des Fonctions 

 Ell.," vol. i., p. 22 ; or Verhulst, § 19, p. 40.) It is no 

 defect, again, that the mechanical applications are omit- 

 ted. These are better studied as they arise, as a part of 

 mechanics rather than of analysis. But as regards geo- 

 metry, we think there has been done either too much or 

 too little. For instance, we have the usual theory of the 

 representation of the arcs of the ellipse and hyperbola 



