April 15, 1920] 



NATURE 



191 



K 



ination of Sex." The peculiar position of Lepi- 

 doptera and Aves with regard to these matters 

 IS emphasised. 



A welcome section on "Germ-cell Determin- 

 ants " gives clearly the main facts which have 

 been ascertained. The author is commendably 

 cautious in his discussion of this interesting 

 subject, and recognises that "although these 

 bodies are evidently strictly correlated with the 

 germ-cells, there is no absolute certainty that they 

 are the cause of the differentiation of germ-cells 

 from body-cells." 



Most cytologists will concur with Prof. Don- 

 caster in his view that the weight of evidence is 

 in favour of the main theory of the individuality 

 of the chromosomes ; the author emphasises the 

 fact that the chromosome itself is in all probability 

 divisible into smaller units, which may have an 

 individuality more fundamental than the chromo- 

 some as a whole, and he suggests that the 

 individuality theory should be extended so as to 

 regard these granules (microsomes) as the funda- 

 mental units. 



The chapter on the mechanism of hereditary 

 transmission introduces a discussion on the most 

 recent work on Drosophila and Abraxas. Prof. 

 Doncaster is here dealing with a subject which he 

 has himself studied specially, and though he treats 

 the question with impartiality, he comes to the 

 conclusion that not only does the behaviour of 

 the chromosomes in the maturation divisions of 

 the germ-cells provide the mechanism required for 

 Mendelian segregation of characters, but also that 

 the work of Morgan on Drosophila carries us a 

 step farther and gives us some idea how the 

 groups of characters may be related to special 

 -chromosomes. The author recognises the diffi- 

 culties with regard to our full acceptance of the 

 theories of Morgan and his colleagues relating to 

 the m^hanism of "coupling" and "crossing 

 over," but states that no other hypothesis which 

 has been brought forward fits the main facts so 

 well. He concludes his book with an 



-essay on the role of the cytoplasm in heredity, and 

 gives a good account of the organ-forming 

 substances. 



Arranged and written as it is, this book is 

 certain to stir up interest in the subject of cyto- 

 logy. By pointing out the perfections and defects 

 of our present-day basic cytological theories and 

 "hypotheses, the author has succeeded in empha- 

 sising the lines along which fruitful research may 

 be followed. We hope that this book will mark 

 the beginning of greater activity among English 

 cytologists. Prof. Doncaster is to be congratu- 

 lated warmly on this excellent work. J. B. G. 

 NO. 2633, VOL. 105] 



Matrices. 



Calcutta : Readership Lectures : 



Determiuoids. By Prof. C. E. 



University of 

 Matrices and 

 Cullis. Vol. ii. Pp. xxiii -1-555. (Cambridge: 

 At the University Press, 1918.) Price 425, net. 



THE history of the imathematical term 

 "matrix" is likely to be very interesting. 

 Its original meaning was an array of symbols 

 (a^n) forming a rectangle of m rows and n 

 columns, out of which determinants were selected 

 by picking out columns (or rows) of the array. 

 A square matrix gives only one associated deter- 

 minant, but a square matrix is not the same thing 

 as a determinant. 



When we change from one set of variables to 

 another by linear relations 



fj- 



:2a, 



ijXi (?=I,2, . 



.7=^2, 



we have an associated matrix (amn), or A, which 

 is square only when the number of variables is 

 the same in each set. In practice this is the most 

 usual case, and it will be simpler to confine our- 

 selves to this for the present. If we take a new 

 set of variables s, such that 



we have a matrix B = (b„„), and by eliminating the 



symbols ji we deduce 



I 

 Zj = SCijXi, 



where the symbols Cy are derived from A, B by 

 a process of "composition," and form a new 

 matrix C. We write C = AB symbolically, and 

 thus start the theory of the multiplication of 

 matrices. There are many analogies with the 

 theory of groups ; for instance, BA must be dis- 

 tinguished from AB, multiplication is associative, 

 and so on. 



Cayley seems to have been the first to develop 

 the theory of square matrices from this point of 

 view (Phil. Trans., vols, cxlviii., clvi., and else- 

 where) ; other English mathematicians, such as 

 Sylvester, Buchheim, and Tait, took up the subject 

 later on. It may be specially noted that H. Smith's 

 memoir on linear indeterminate equations and con- 

 gruences contains a great deal of the fundamental 

 theory of matrices, both square and rectangular. 

 In particular, there is a complete and, we believe, 

 original statement of the existence and properties 

 of the elementary factors of a determinant the 

 elements of which are ordinary integers. Weier- 

 strass, Kronecker, and Frobenius, especially the 

 last-named, have made important contributions to 

 the subject. 



It will be seen that a matrix is now not merely a 

 scheme of symbols used to specify a set of deter^ 



