46: 



NA TURE 



[September 6, 1906' 



channels and the reclamation of land in consequence 

 have resulted. It is in this way that the lake occupy- 

 ing the depression of the Fayum has been diminished 

 in area. 



The floods in the Kile Delta of which records have 

 been kept, that are trustworthy for the past 175 years 

 at least, have been critically studied by Captain Lyons 

 with the view of discovering the determining causes 

 of their variations. While no regular periodic alter- 

 nations of high and low floods can be detected by the 

 study of these records, their dependence on the rain- 

 fall and the distribution of atmospheric pressure in the 

 highlands of Equatorial Africa is very apparent. 

 Thero is reason to believe that more numerous, 

 systematic, and complete meteorological observations 

 in the districts outside Egypt may enable us, in the 

 end, to predict from month to month the probable 

 fluctuations of the annual Nile flood. 



The sipace at our command has only permitled the 

 notice of a few of fhe more salient features of this 

 very interesting volume. In conclusion, we must 

 congratulate Captain Lyons and the Egyptian Govern- 

 ment upon the great amount of valuable work which 

 has been accomplished and is still in progress. A 

 word of praise must also be added on the excellent 

 typography of the volume, and the admirable plates 

 with which it is illustrated. J. W. J. 



THE HISTORY OF DETERMINANTS. 



The Theory of Determinants' in the Historical Order 

 of Development. P,'u-t i. Second edition. General 

 Determinants up to 1841. P^rt ii. Special Deter- 

 minants up to 1841. By Dr. T. Muir, C.M.G., 

 F.R.S. Pp. xii-l-492. (London: Macmilhm and 

 Co., Ltd., 1906.) Price 17s. net. 



MATHEMATICAL history of the riglic sort is 

 much more than a mere bibliography, and in 

 some respects is more valuable than a treatise on the 

 subject with which it deals. It helps us to see how 

 mathematical ideas originate, and how, as they be- 

 come familiar, the symbolism by which they are 

 expressed becomes compact and appropriate. This 

 is especially the case with determinants, because 

 a determinant is essentially a comprehensive symbol, 

 and it would perhaps be more proper to speak of 

 the calculus than of the theory of determinants. 

 It may seem strange, at first sight, to find a history 

 so large as this dealing with a subject so limited ; 

 hut no one can complain that the author is either 

 difl'use or irrelevant, and his work may be praised 

 without restriction as a model of its kind. 



It is unnecessary to say much of the first part, 

 which is mainly a reprint of the volume which 

 .Tppeared in 1900. Dr. Muir has written a new in- 

 troduction, and added a few additional notices. Two 

 things cannot fail to strike the reader of this part. 

 The first is the great supremacy of Cauchy and 

 Jacobi in everythin.g relating to choice of notation and 

 clearness of statement ; the other is the great and 

 long unrecognised ability of Schweins. Schweins, in 

 a way, brought this fate upon himself; his style is 

 NO. 1923, VOL. 74] 



A 



heavy, and his notation cumbrous in the extreme, but 

 his contributions to the subject are of great value and 

 generality, although they attracted no notice for many 

 years, and were re-discovered by others. Unfortu- 

 nately, they are expressed in such a repulsive notation 

 that no one but an enthusiast would read his works, 

 and the student will feel very grateful to Dr. Muir 

 for his analysis of them. Part of this analysis, in 

 some ways the most interesting, is given on pp. 311- 

 322 ; this, and the subsequent section on a paper of 

 .Sylvester's, deserve careful reading, because, as Dr. 

 Muir points out, Schweins gives some results on 

 alternants which even now are not familiar, and 

 Sylvester makes some hasty statements which, as 

 they stand, appear to be incorrect, but which, if cor- 

 rected, or rightly interpreted, might lead to important 

 formulae. 



It should be noted that on p. 323 the determinant 

 is misprinted, a, a", &c., being put for a,, o^, &c. 

 Moreover, it is not explained so clearly as it should 

 be that (n''=t>,.; while the law (?,■.««=«,• + « 's nof used. 

 The right statement is t("'' ■ "") = f(«'"*^")='J'i- + « ; while 

 i(a'')C{a') — a,.ag. Readers of Sylvester's papers must 

 be careful to distinguish this C from the square of 

 the operator C-- It niay be noticed, in passing 

 that these generalised alternants present themselves in 

 the theory of numbers, both when the elements are 

 roots of unity and also when they are not, so that 

 further knowledge of their properties is desirable, and 

 the suggestion made (p. 325) that Sylvester's results 

 are true when the elements are periodic deserves 

 further examination. 



Considerable space is given to functional and ortho- 

 gonal determinants, and here, of course, Jacobi 

 receives most attention. The results are now so 

 familiar that it requires some effort of imagination to 

 realise the gain in working power which has resulted 

 from Jacobi 's investigations. In this connection 

 attention may be directed to an odd reinark on p. 297. 

 Speaking of one of Jacobi "s papers. Dr. Muir says : — 

 " The only thing worth noticing is the curious cubic 

 equation . . . "; this "curious " equation is nothing 

 more nor less than the reducing cubic for two ternary 

 quadratic forms, in the exact notation of Salmon's 

 " Conies " ! .And Dr. Muir even takes the trouble to 

 express the invariants e, ©' in the forms 



Aa' + Bb' + 



A'a + B'b + 



&c.. 



as if this were a quL'e novel idea. 



Returning for a moment to alternants and their 

 applications, attention may be directed to the work of 

 Jacobi and Cauchy on the expansions of rational func- 

 tions of several variables (pp. 331-345). This is 

 important in the theory of functions, in that of 

 algebraic forms, and in that of partitions. In some 

 ways it deserves further investigation ; in various 

 applications the expansions have to be infinite series, 

 and the question of convergericy has to be faced, even 

 when the series are used for establishing formal equi- 

 valences ; this is a curious case of formal and arith- 

 metical alg'ebra each marching, so to speak, on the 

 other's dom-iin. 



