148 



NATURE 



[December 19, 1895 



Another very interesting chapter follows. This contains 

 Hilbert's proof of Gordan's celebrated theorem, that the 

 number of irreducible concomitants is finite. Compared 

 with Gordan's original proof, this is simplicity itself; and 

 it is unlikely that the demonstration can be essentially 

 improved upon in this respect, although no doubt some 

 simplification in detail may be effected. 



Chapters x. and xi. are also well brought up to date. 

 They deal with protomorphs and perpetuants, and the 

 connection of seminvariants with non-unitary symmetric 

 functions. It is needless to say that they are based 

 principally on the researches of MacMahon and Ham- 

 mond. The deduction of the annihilator of non-unitary 

 symmetric functions of the quantic 



^x + -i- X' 



seems rather artificial, as it is made to depend upon the 

 transformation 



But this is a small matter, and the chapters are full of 

 interest. One remarkable novelty is a differential operator 

 which annihilates any rational integral function whatever 

 of the coefficients of a finite quantic. Here is, indeed, a 

 universal solvent. It should be added that the examples 

 at the end of chapter xi. give a synopsis of Stroh's veri- 

 fication of MacMahon's brilliant conjecture that the 

 generating function for perpetuants of degree i is 



(I -X'){1 -X3) 



(I 



{^>^) 



, The remaining chapters (xii.-xvi.) treat of canonical 

 forms, the binary quintic and sextic, systems of binary 

 quantics, orthogonal invariants, and the ternary quadratic 

 and cubic. The chapter on the quintic and sextic does not 

 go into detail, but gives complete lists of the concomitants, 

 and in particular the explicit forms (supplied by Mr. 

 Hammond) for the quintic (a, b, o, o, e^f) (.v,}'f. The 

 other chapters do not seem to call for special remark ; 

 suffice it to say that they maintain the high standard of 

 those which precede them. 



Prof Elliott states in his preface that the book is an 

 expansion of a course of lectures delivered annually for 

 some years past at Oxford. To this fact, no doubt, may 

 be attributed, in some measure, the lucidity and sym- 

 metry of the treatise. Another good feature, perhaps 

 due to the same cause, is the occasional statement of 

 what a theorem does «<?/ imply. To the well-informed 

 reader this may seem superfluous, but it is by no means 

 so in the case of a learner, who not infrequently reads 

 into a theorem a degree of generality which it does not 

 reaHy contain. 



To return to the symbolical method of Clebsch and 

 Aronhold. Prof Elliott admits that an English work on 

 this calculus is a desideratum ; will he not be persuaded 

 to supply this want himself? It would be a great boon 

 to have an English book something after the kind of the 

 Clebsch- Lindemann " Geometric," including, at least, the 

 theory of plane quadratic and cubic curves, and of surfaces 

 of the second order, with perhaps an introduction to the 

 theory of cubic surfaces. This is, no doubt, a heavy task, 

 but it is well worth attempting ; the theory of forms is 

 NO. 1364, VOL. 53] 



infinitely more interesting in its geometrical applications 

 than as a mere branch of analysis, and it is here, above 

 all, that the power of the symbolic method shows itself. 

 Such a work would do much to avert the danger of 

 divorcing the theory of forms from analytical geometry ; 

 a danger which is encouraged by the present regulations 

 of the mathematical tripos, which place these cognate 

 subjects in two different divisions. G. B. M. 



SURFACE-COLOURS. 

 Die Oberfldchen- oder Schiller- Farben. Von Dr. B. 

 Walter, i vol., with 8 woodcuts and i plate. Pp. vi. 

 -I- 122. (Braunschweig: F. Vieweg und Sohn, 1895.) 



THIS work is primarily addressed to zoologists, 

 mineralogists, and chemists, appealing in only a 

 subordinate measure to physicists. On this account 

 the mathematical developments most desirable for the 

 physicist are reserved for appendices, while the text 

 itself contains only such matter as is vital to the theory 

 of surface-colours, together with very simple and well- 

 established formulae given without proof. 



The importance to the first-named classes of an ac- 

 quaintance with the physical basis of these colours 

 immediately appears, says the author, from the facts that, 

 on the one hand, to this class of colours belong the tints 

 of many butterflies and birds, and also those of a series 

 of crystals exhibiting the most gorgeous natural phe- 

 nomena ; and on the other hand, the technologist, if 

 he desires artistically to imitate these colours, must 

 naturally, first of all, obtain a true insight into their 

 manner of production. Now, although this treatise con- 

 tains no startling additions to our physical knowledge 

 of surface-colours, it may yet be expected to render a 

 most acceptable service to this branch of physics, since 

 in many minds there still linger hazy, or even discordant,, 

 conceptions of these colours, and until now no work 

 seems to have appeared devoting to the subject even any 

 approach to an exhaustive treatment. 



Of the experiments, which form the basis of the cal- 

 culations and statements contained in this book, those 

 which are new have been carried out by the author in 

 the State Physical Laboratory at Hamburg. 



The first chapter is a brief introduction to the subject 

 of the work. The second and third chapters treat of 

 the surface-colours of colourless materials and of metals 

 respectively. In the fourth chapter, embracing about a 

 third of the entire treatise, the author discusses the 

 dichroic substances proper ; solid fuchsine and diamond 

 green, also solutions of these, and fluorescein solution 

 being specially dealt with. This chapter is unusually 

 rich in experimental results. It is pointed out that the 

 body-colour and the surface-colour are only approximately 

 complementary, and are not exactly so, as stated in Haid- 

 inger's law. We have also here the following important 

 statement : " The coefficient of reflexion of a particular 

 ray from a given substance depends not only upon the 

 absorption coefficient of the substance for that ray, but 

 also upon its refractive index for the ray in question, 

 the relative importance of these two factors varying with 

 circumstances, so that in the case of the feebly-absorbed 

 rays of coloured substances the refractive index is prac- 



