August 4, 1904] 



NA TURE 



115 



collected. In addition to those on niyriapods and 

 arachnids, the first of the insect volumes includes 

 papers on Alaskan Coleoptera, Lepidoptera, Neuro- 

 ptera, Orthoptera, Homoptera, and various minor 

 groups. Special interest attaches to the chapter by 

 Prof. Kincaid on the metamorphoses of .Alaskan 

 Coleoptera, in the course of which the author points 

 out that the prevalent idea as to the extreme difficulty 

 of rearing adult beetles from their larval condition is 

 to a great e.xtent founded on error. 



The second of the two volumes on insects (ix.) is 

 devoted to the Diptera and Hymenoptera, the article 

 on the former being written by Mr. W. Coquillett, and 

 the one on the latter group by Mr. W. H. Ashmead. 

 In addition to these are three minor papers on certain 

 sections of the aforesaid groups. \ sample of the 

 excellent results of Prof. Kincaid 's energetic collect- 

 ing is afforded by the case of the Hymenoptera, in 

 which group less than thirty species were known from 

 Alaska previous to the expedition, while the number 

 now recorded is no less than 335, 201 of these being 

 regarded as new to science. Of Diptera, 2423 speci- 

 mens, representing 276 species, were collected, out of 

 which one genus and 63 species are described as new. 



As regards the volume on crustaceans, the great 

 bulk of this is occupied by Miss Rathbun's elaborate 

 and exhaustive memoir on the decapod section. Miss 

 Richardson contributing a short account of the 

 isopods, while Messrs. Holmes and Cole are severally 

 responsible for the amphipods and pycnogohids, or 

 sea-spiders. Miss Rathbun has treated her portion of 

 this extensive subject from a very broad standpoint, 

 discussing the crabs and shrimps not only of the 

 .\laskan seas, but of the western coast of America 

 generally, from the Arctic Circle to southern Cali- 

 fornia. The decapod fauna of the North Pacific has 

 proved very rich in individuals, if not in species. 

 Among the more abundant types are the hermit-crabs, 

 of which many species have local centres of distribu- 

 tion, where they attain their maximum development, 

 both as regards size and numbers. In certain locali- 

 ties this crowding of crustacean life has been specially 

 favourable to the development of parasitism. Tlie 

 decapods form the staple food of many kinds of fishes, 

 and certain species are commonly used by fishermen 

 as bait, or caught for the table. In many cases the 

 northern limits of the species are determined by the 

 winter line of floating ice in Bering Sea. 



As regards the other groups, it must suffice to say 

 that while Mr. Holmes restricts himself to the amphi- 

 pods collected during the expedition, the isopods and 

 pycnogonids of the whole western coast, from Alaska 

 to California, are discussed. It is perhaps this vari- 

 ation in the mode in which the different groups are 

 treated that constitutes the main ground for criticism 

 in regard to the general plan of this magnificent and 

 valuable work. Both editor and contributors are to 

 be congratulated upon the results of their labours, so 

 far as these are at present before the public, while the 

 thanks of the scientific world are especially due to 

 Mr. Harriman, as the generous provider of the means 

 whereby this important addition to knowledge has 

 been rendered possible. R. \_. 



NO. 1 8 14 VOL. 70] 



THE THEORY OF DETERMINANTS. 

 The Theory of Determinants. By R. F. Scott, M.A. 

 Second edition. Revised by G. B. Mathews, M.A., 

 F.R.S. Pp. xi + 288. (Cambridge: University 

 Press, 1904.) Price qs. net. 



THIS well known treatise has been revised and 

 enlarged in several respects. For instance 

 (chapter xi.), the theory of linear equations is more 

 complete than in the first edition, and Bezout's method 

 of elimination is explained, as well as Sylvester's. 



An introductory chapter has been inserted, contain- 

 ing an elementary account of three-rowed deter- 

 minants ; this should prove a considerable help to 

 beginners. As a whole, the new edition is probably 

 easier reading than the first ; but even now the style 

 seems rather too condensed for the average reader, and 

 illustrations of general theorems by special cases are 

 somewhat scarce. 



A chapter (x.) on infinite determinants has been 

 added ; this appears to be based on the work of 

 von Koch and Cazzaniga, but as some investigations 

 have been abbreviated, occasional difficulties may be 

 encountered at a first reading. Thus von Koch's proof 

 that a normal determinant converges (art. 5) would 

 be clearer if reproduced in full, and the convergence- 

 test employed here ' might be explained at greater 

 length. The investigations of arts. 6 and 10 assume 

 that certain infinite sequences (a^j. and C,jt) have upper 

 limits ; von Koch establishes this property by com- 

 parison with infinite products. 



Semi-normal determinants are defined, in art. 11,. 

 after Cazzaniga; von Koch's definition would give a 

 more elegant form to the theory without loss of 

 generality. The two rules for multiplying semi- 

 normals are stated in art. 12 ; but C is not proved to- 

 be equal to AB, and the statement (p. 128) " the series 

 Cit is absolutely convergent " must not be taken to 

 refer to 2<",j;. Some examples like Cazzaniga 's would 

 emphasise the contrast between these rules and the 

 four rule's of art. 10 (for multiplying normal deter- 

 minants). 



On several grounds it is regrettable that chapter x. 

 is not more complete. No proof is given that the 

 value of a normal determinant is the same, wherever 

 the origin may be taken on the principal diagonal;, 

 and various analogies with finite determinants are 

 omitted. 



A new chapter (vii.) has been inserted, containing 

 the simpler theorems on Elementartheiler of deter- 

 minants ; this term is translated literally elementary- 

 divisors, although several English writers have used 

 invariant-factors as the equivalent. The treatment 

 follows Dr. Muth's bcx)k very closely; we have ex- 

 plained elsewhere (Bulletin Amer. Math. Soc, vol. vii. 

 p. 308) that some changes in Dr. Muth's order might 

 make the work more readable. But, in default of any 

 English text-book, we must welcome this chapter as 

 a useful introduction to the subject. 



Frobenius's calculus of bilinear forms is explained 



1 '* A sequence An converges if lim (A„+p-Ajt)=o, for all positive- 



