December 19, 1895] 



NATURE 



147 



are also certain points to which attention must be drawn, 

 in view of future editions of the book. 



When dealing with plasmolysis (p. yj), the author 

 omits to state that the animal or vegetable cells used 

 must contain living protoplasm, and the reader is led to 

 infer that artificially-coloured, instead of naturally- 

 coloured cells are employed for plasmolytic observations. 

 Although certain stains are known which are not 

 immediately fatal to living cells, there is no record of 

 their use in plasmolytic experiments. It is also made 

 to appear that red blood corpuscles contain a semi- 

 permeable membrane, despite the conclusive observ- 

 ations of Hamburger to the contrary. In connection 

 tvith this subject, it is misleading to state (p. 38) that 

 De Vries " established the most important generalisation " 

 that solutions of the sime molecular concentration 

 are isotonic, for inasmuch as by far the greater number 

 of his solutions were electrolytic, his results clearly con- 

 tradict this statement. On p. 34, the credit of preparing 

 semi-permeable membranes is given to Pfeffer, whereas 

 M. Traube first described their preparation and proper- 

 ties. As regards the more general treatment of the first 

 section, it is noteworthy that although the solubility of 

 mixed substances is to some extent discussed, no notice 

 is taken of the work of Roozeboom, and Gibbs' phase- 

 rules, which apply to all cases of heterogeneous equili- 

 brium in solution, are not even mentioned. 



In the section on electrolysis, some inkling might have 

 been given of the wide field opened up for the verification 

 of the ionic hypothesis by its application to the operations 

 of analytical chemistry. Among smaller points, it may be 

 noted that, on p. 128, potassium platinichloride should 

 be sodium platinichloride, and in a somewhat vague 

 paragraph, on p. 164, we read that the introduction of 

 oxygen, sulphur, or a halogen, which raises the affinity of 

 a weak acid, "has no effect on theaffinity of these strong 

 acids." Since the strong acids quoted by the author are 

 hydrochloric, nitric, &c., the student may be pardoned if 

 he is puzzled to understand how the introduction is to be 

 brought about, or what acids would result if it were possible. 

 A novel feature in a book of this kind is an attempt 

 made by the author to reconcile the Hydrate Theory with 

 the Newer Theory of solutions. Of course it has all along 

 been apparent that the latter does not preclude combina- 

 tion between solvent and dissolved substance. What the 

 upholders of the newer theory assert, however, is that at 

 the present time there is no definite evidence that, in 

 general, such combination exists. An attempt to reconcile 

 the two views should therefore involve a careful study of 

 the experimental data in favour of combination. It is for 

 this reason unfortunate that the author gives but a very 

 brief statement of the results of the extensive work of 

 Pickering in this field. 



As an appendix to the book is given part of the list of 

 the conductivity, migration, and fluidity data of solutions 

 compiled by Fitzpatrick for the British Association Report 

 of 1893. Fo'' the sake of chemical readers it is to be 

 regretted that most of Ostwald's observations on the 

 conductivity of organic substances have been omitted, 

 since it is in the case of such substances that the close 

 connection between the electrolytic properties of solutions 

 and the chemical nature of the dissolved substances can 

 be most conveniently traced. J. W. RODGER. 



NO. 1304, VOL. 53] 



THE THEORY OF ALGEBRAIC FORMS. 

 An Introduction to the Algebra of Quantics. By E. B. 

 Elliott, M.A., F.R.S. Pp. xiv. + 424. (Oxford : 

 Clarendon Press, 1895.) 



THE history of the theory of algebraic forms gives a 

 striking example of the fact that the germ of a 

 mathematical doctrine may remain dormant for a long 

 period, and then suddenly develop in a most surprising 

 way. The principles of the calculus of forms are to be 

 found in the arithmetical works of Lagrange, Gauss, and 

 Eisenstein ; but the great expansion of the theory, with 

 which we are now so familiar, practically dates from the 

 publication of the papers of Boole, Cayley, and Sylvester, 

 about fifty years ago. 



It is well known that the theory of forms has advanced 

 upon two distinct lines : one method being derived 

 mainly from the differential equation of sources, supple- 

 mented by generating functions and the theory of 

 equations ; the other, from the symbolical representation 

 of a quantic, invented by Aronhold, and applied with 

 such power by Clebsch and Gordan. Until quite lately, 

 the symbolical method might not unjustly claim to be 

 superior in respect of organic unity, as it must still be 

 admitted to be in compactness and geometrical sugges- 

 tiveness ; but the other method has now undergone a 

 remarkable transformation at the hands of Hammond, 

 MacMahon, Hilbert, and others, and has led to results of 

 the highest interest and value, which the symbolical 

 calculus could not easily or naturally supply. 



With the exception of three pages, devoted principally 

 to Cayley's hyperdeterminant notation. Prof. Elliott does 

 not refer to the symbolical method. With his reasons 

 for not using it we must reluctantly acquiesce. It is 

 quite true, as he says, that a mere outline of the method 

 would have been worse than useless ; and by omitting it 

 altogether, he has been enabled to give a very lucid and 

 thorough account of the subject from one consistent point 

 of view, without that excessive condensation which is so 

 often a defect rather than a merit. 



It is not necessary to say much of the earlier chapters, 

 except that, like the rest of the book, they are very clear 

 and pleasant to read ; in particular, the proof that every 

 covariant of a covariant is a covariant of the original 

 form is easier to follow than that given by Salmon. It 

 is when we come to chapters vi. and vii., which deal with 

 seminvariants and their annihilators, that the influence of 

 recent discoveries begins to be felt. Thus the notions of 

 excess and extent are introduced, and the annihilators of 

 invariants and covariants of systems of quantics are 

 indicated. 



Chapter viii. discusses generating functions, and is a 

 very good introduction to this part of the subject. It 

 does not profess to be exhaustive ; and it is perhaps as 

 well that the author has refrained from giving the de- 

 tailed reduction of the generating functions for forms 

 higher than the quartic. This would have taken up a 

 good deal of space ; and the full discussion for the lower 

 forms, which is given, is quite enough to illustrate the 

 eeneral procedure. The results for the quintic are also 

 stated, and references are given to the memoirs of 

 Sylvester and Franklin, which ought to be easily under- 

 stood by any one who has mastered this chapter. 



