Sept. 1 6, 1880] 



NATURE 



459 



on the subject was the work of an Englishman, now the 

 distinguished president of the Royal Society. Dr. Spottis- 

 woode's "Elementary Theorems Relating to Determin- 

 ants" appeared in 1851 (4to, pp. viii. + 63, London, 

 Longmans), and, as a pioneer work, was eminently suc- 

 cessful ; at the honouring request of the editor of Crellc 

 it was republished, with additions, in that well-known 

 journal three or four years later (vol. li. pp. 209-271, 

 328-381). After considerable intervals came Dodgson's 

 " Elementary Treatise " (London, Macmillan, 1867) and 

 a pamphlet by Wright ; and here, until quite recently, 

 the list ended. The chapters on the subject by Tod- 

 hunter and others belong to a different category, but 

 deserve to be mentioned, as it is doubtless in part owing 

 to their existence that separate treatises have been so 

 rare. 



In view of the dearth referred to, he would be a very 

 captious critic indeed who would not gladly welcome the 

 handsome volume whose title is given at the head of this 

 notice. In form and general outward appearance it re- 

 sembles Part I. of Thomson and Tait's "Elements of 

 Natural Philosophy," and extends to about 250 pages. 

 The matter is arranged under fourteen chapters, the first 

 seven being meant to deal with determinants in them- 

 selves, the last seven with the so-called applications ; the 

 line of separation, however, is not very well maintained. 



In the introductory chapter we have the usual account 

 of permutations, inversions of order, &c., and the usual 

 definition of a determinant; but this is followed by some- 

 thing less famihar, viz., a page or two of exposition 

 regarding Grassmann's "alternate units " or "polar ele- 

 ments," and by the establishment of the theorem that a 

 determinant is expressible as a product of alternate num. 

 bers linear in the elements. ' The constant use afterwards 

 made of this theorem— if theorem we can call that which 

 is but a symbolical expression of the ordinary definition 

 of a determinant— is the distinguishing feature of Mr. 

 Scott's mode of treating the subject. There may be room 

 for doubt whether the study of determinants is thus, as 

 he says, much simplified— the example of § 20, p. 15, is 

 not a happy introductory instance of such simplification— 

 and it may certainly be questioned whether beginners 

 should have the subject at first presented to them in this 

 way ; but undeniably a freshness is thereby imparted to 

 the book, which wiU make it pleasant reading to those 

 who already know something of the matters in hand. 

 Chapters II. to V., on " General Properties of Deter- 

 minants," "The Minors and the Expansion of a Determi- 

 nant," " Multiplication of Determinants," and " Deter- 

 minants of Compound Systems," contain proofs and 

 illustrations of W()jY of the well-known general theorems. 

 One might, however, fairly expect so large a work as the 

 present to be more complete in this respect ; the omissions 

 for example, of Sylvester's beautiful theorem expressing 

 the product of two determinants as a sum of like products 

 is not easily excusable. Chapter VI., on " Determinants 

 of Special Forms," is good, and the same may be said of 

 the next, which treats of " Determinants with Multiple 

 Suffi.xes." Belonging to the first part, although included 

 under "Applications," are Chapters IX. and XII. The 



one concerns what are called " Rational Functional De- 

 terminants," but which might be more fitly designated as 

 " Alternants "—to use one of Sylvester's happiest coin- 

 ages ; the other concerns " Determinants of Functions of 

 the same Variable," a title again which is anything but 

 sufficiently discriminative. Both chapters are fresh and 

 interesting. To the borderland between the two parts 

 may be assigned Chapter X., on " Jacobians and Hes- 

 sians;" then there are chapters (VIII., XI., XIII.)dealing 

 with the applications to three departments of Analysis, 

 viz., Theory of Equations, Theory of Ouadrics, and Con- 

 tinued Fractions ; and, lastly, there is a very readable 

 chapter (XIV.) on " The Applications to Geometry." 



No exercises for the student are given under the 

 individual chapters, but a considerable collection is placed 

 towards the end. 



Following this is a " List of Memoirs and Works 

 Relating to Determinants," the arrangement being alpha- 

 betical according to authors' names. Mr. Scott acknow- 

 ledges the incompleteness of the list ; but, -all the same. 

 one cannot help expressing regret that such an excellent 

 opportunity of publishing an exhaustive, or tolerably ex- 

 haustive, bibliography of determinants was lost. Had the 

 list been a judicious selection, there would have been less 

 cause for regret, but not rarely the worthless are taken 

 and the good left out. Cauchy, who in a sense laid the 

 foundations of the whole subject, is not once mentioned ; 

 Grassmann, who laid the foundation of Mr. Scott's method, 

 is not included ; Nagelsbach's name occurs, but his most 

 important paper is omitted ; and many more such 

 instances might be cited. 



Mr. Scott has given us a very acceptable addition to 

 our mathematical text-books : a little more of the con- 

 scientious labour he has shown would have produced a 

 work still more worthy of the press which has issued it. 



.g.\d <• y = («,, + 1,,^ + «3) (^,_ + , 



1^ '•■ k, 

 where ejEj.j = i, and . 

 algebra, except that £,£ 



ire symbols subject to the 1 

 -,'and therefore c.^ = o. 



+ /n) (f^f, •!■/«, -I- ^£3), 

 of ordinary 



LETTERS TO THE EDITOR 



[The Editor does not hold himself responsible for opinions expressed 

 by his correspondents. N'cither can he undertake to return, or 

 to conespond with the writers of, rejected manuscripts. No 

 notice is taken of anonymotts eomtmtnieations.'\ 

 [ Tlie Editor urgently requests correspondents to keep their letters as 

 short as possible. The pressure on h is space is so great that it 

 is impossible otherwise to ensure the appearance even of cotn- 

 munications containing interesting and novel facts.} 

 The Stone in the Nest of the Swallow 

 The name of swallow's stone was preserved in France even to 

 our days, for Dr. Patrin, a member of the French Institute and 

 of the Academy of Sciences in .St. Petersburg, wrote in the 

 "Dictionary of Natural History," Paris, Deterville iSol (V° 

 Agate), as follows : — > J v 



" On troiive dans les ruisse.iux des environs de Sassenage en 

 Dauphine de tres petites Calcedoines ou Agates de forme lenti- 

 culaire qu on a nommees pierres de ChJlidoine, parce qu'elles ont 

 quelque ressemblance avec les semences de cette ^l&nte,' pierres 

 d'/urondelle parcequ'on en a trouve dans I'estomac de ces 

 oiseaux." 



Of course this naturalist did not try to throw light on the 

 legend, or to explain the confusion made by some authors between 

 the respective skiU of the eagles in geology, and of the swallows 

 m botany, which Phile, in his "Remedies Against Sortileges," 

 clearly sets out in the following verses : 



(p6opas Se TT^pfi Taj yovas imfpTepas 

 eij TTiD KaXiav aeris Kpvtpas \leoi/ 

 ws J) x«'^-'5w;' ToC creKlmv rriv K6fnji' 

 'E^a\fiaira Se tov Tpaxii\ov riii' \i0ov 

 K'jovcra ■yvvTj KepZati/u (an' Tiffpetpos. 

 " .\ ;-tone which the eagle conceals in her nest (aery) preserves 



