December 31, 19 14] 



NATURE 



473 



TWO MATHEMATICAL COURSES. 



(ij The Theory of Numbers. By Prof. R. D. 

 Carmichael. Pp. 94. (Mathematical Mono- 

 graphs, No. 13.) (New York : John Wiley and 

 Sons, Inc.; London: Chapman and Hall, Ltd., 

 1914.) Price 45. 6d. net. 



(2) Lectures Introductory to the Theory of Func- 

 tions of Two Complex Variables. Delivered to 

 the University of Calcutta during- January and 

 February, 1913, by Prof. A. R. Forsyth. Pp. 

 xvi + 281. (Cambridge: University Press, 

 1914.) Price 105. net. 



THESE works are mainly interesting as ex- 

 amples of the trend of instruction given to 

 university students who are just beginning to 

 specialise. One is based upon two years' teaching 

 in Indiana University; the other is the substance 

 of a course delivered, by invitation, at the Uni- 

 versity of Calcutta, with, presumably, a number 

 of Indian students in the audience. Both courses 

 agree in the endeavour to trace the main outlines 

 of the argument, to give really illustrative ex- 

 amples, and to avoid excessive detail on parti- 

 cular points of minor importance. 



(1) Prof. Carmichael does not advance beyond 

 well-beaten and classical ground, so that he has 

 little opportunity of introducing really modern 

 ideas. All but one of his six chapters deal with 

 such things as ordinary factorisation of integers, 

 elementar}' congruences, Fermat's theorem, and 

 primitive roots. Chapter vi. is a miscellany in 

 which there is a statement, without proof, of the 

 law of quadratic reciprocity for two odd primers ; 

 a very brief account (with references, fortunately) 

 of some special Galois imaginaries, and some in- 

 teresting samples of the true Diophantine analy- 

 sis. Some of the harder examples are novel and 

 interesting, e.g. "The number of divisions to be 

 effected in finding the G.C.M. of two numbers by 

 the Euclidean algorithm does not exceed five 

 times the number of digits in the smaller number 

 (supposed written in the decimal scale)." 



W^ithout advising any improper and premature 

 specialisation, it may be fairly urged that some 

 of the work contained in this course might be 

 done at school; the G.C.M. theory, and that of 

 linear congruences, at any rate. Familiarity with 

 the congruence notation is so important that its 

 introduction ought not to be deferred. 



(2) Prof. Forsyth has undertaken the difficult 

 task of giving an outline of the known theory of 

 functions of two independent (complex) variables, 

 together with an account of what he calls "triple 

 theta-f unctions." Everyone who has studied this 

 theory at all is aware that it suggests theorems 

 similar to those of Gauss, Green, Cauchy, etc., 



NO. 2357, VOL. 94] 



for functions of one variable, but that it is very 

 difficult to prove them in an analogous, so to 

 speak, geometrical way, mainly because while a 

 plane or a sphere gives us a graphic picture of the 

 field of one complex variable, we cannot at pre- 

 sent realise a convenient image of the field of two 

 independent complex variables. It is easy enough 

 to make images of a sort; for instance, take two 

 planes, plot off the independent variables, x, y, in 

 the usual way on each, and now associate with a 

 given pair {x', y'), the line joining the point which 

 is the image of .x' to that which is the image of 

 y'. An obvious objection to this is that the meet 

 of the planes does not correspond to a unique pair 

 x', y', and there are other inconveniences con- 

 nected with the difficulty of visualising linear com- 

 plexes and congruences. After reviewing various 

 proposals, Prof. Forsyth concludes that the only 

 practicable way at present is the purely analytical 

 one, following the methods of Weierstrass and his 

 school. It is to be hoped that this is not the 

 final word on the question ; at any rate, there are 

 papers by Picard, Appell, and Poincare which 

 ought to stimulate those whose ideas naturally 

 clothe themselves in geometrical forms. 



Features of the course which should be noted 

 are : (i) the introduction of two dependent vari- 

 ables, now and then, as functions of two indepen- 

 dent variables ; the use of this is analogous to 

 that of one-one transformations of plane curves ; 

 (2) in the chapter on integrals, where we have 

 two algebraic functions introduced, so that the 

 independent variables are arbitran,- ; (3) the theon.- 

 of the so-called "triple theta-f unctions. " The 

 characteristic equations of these functions are 

 given in the form : 

 ^(^+r, 2') = ^(s,s'+i) = d(s,3') 



^(z ^ fJL, S + ix')= exp.' — 2iri{2Z + 2S') - 



2-/(u + /x')}. d(2,2'). 



Hence the author derives double Fourier expan- 

 sions for the functions, which fall into a co-ordin- 

 ate set of sixteen; various tables and formulae 

 relating to them are given. The course con- 

 cludes with a sketch of the theory of quadruply 

 periodic (double) theta-functions, and their alge- 

 braic relations. 



It seems to us that this course is not quite so 

 well-proportioned or up-to-date as that of Prof. 

 Osgood, recently published, on the same subject; 

 but the difference of object and of audience must 

 be allowed for. At any rate, we have a useful 

 guide to the work of Weierstrass and Picard, a 

 certain amount of new, although not very funda- 

 mental, theor}' ; some instructive and original 

 examples ; and, it need scarcely be said, an elegant 

 analytical presentation of the subjects treated. 



G. B. M. 



