500 Notices respecting New Boohs. 



by vectors, of which real quantities represented by segments of the 

 axis and " imaginary " by vectors perpendicular to that line are 

 particular cases. Perhaps more unity would have been imparted 

 to the terminology by the substitution of "unreal" for "ima- 

 ginary;" then ''real" would have expressed the complex of which 

 the component perpendicular to the axis vanishes ; " unreal," that 

 the real component of which is zero. The abbreviation cis. 0, to be 

 read sector of d, is adopted for the complex cos Q-\-i sin 0. The 

 associative, commutative, and distributive laws are then proved 

 for complex magnitudes. All this has, of course, been founded on 

 Argand's construction, which, however, is only formally explained 

 hereafter. 



In the second section of this Chap. III. the Exponentials and 

 Logarithms of complex quantities are dealt with by means of a 

 diagram ; the theory as adopted having been the subject-matter of 

 a paper by the author read before the New York Mathematical 

 Society in October 1891, and subsequently published in vol. xiv. of 

 the Am. Journ. of Math. Without the diagram, a description of 

 this theory would be hardly intelligible. Just as the vanishing of 

 an angle causes the complex to degenerate into a real, so in the 

 diagram referred to the vanishing of an arbitrary angle in it 

 changes the Exponentials and Logarithms of complex into those of 

 ordinary magnitudes. The result of these discussions leads to the 

 definition of " an Algebra ; " viz. since the aggregate of all com- 

 plexes operating howsoever on one another form a " closed group," 

 i. e. evolve only complexes, such a group forms an Algebra. 

 Eeference is made to Peirce and Cayley. 



The chapter concludes with proofs that every magnitude can be 

 expressed, with an error less than any assignable, in terms of an 

 assumed unit by a rational number. 



Chap. IV. treats of the Cyclic and Hyperbolic Functions (direct 

 and inverse) of complexes, defined with respect to a complex 

 modulus (k), and hence called " modo-cyclic " functions, having a 

 period 2%kt: or in-tr, and of the inverses of these. 



Chap. V. explains the one to one correspondence of points on a 

 plane and on a sphere, having its centre in the plane, through rays 

 drawn from one end of the normal diameter, and the transforma- 

 tion of the planar complex into the tri-dimensional sphero-surface 

 complex (Cayley, Klein). The rest of this chapter on " Graphical 

 Transformation " is occupied with some particular cases of " Ortho- 

 morphosis" (Cayley). In it occurs the only misprint which has 

 been noticed, of a symbol B" 7 for B w o. The typography is highly 

 creditable to the San Francisco press — is, in fact, dainty of its 

 kind. 



In the concluding chapter, " Properties of Polynomials " with 

 complex coefficients are proved, after a preliminary page on the 

 evaluation of the n-nth. roots of a complex quantity ; this, as every 

 other branch of the matter of the book, being illustrated by a 

 collection of " Agenda " or Examples, on which the student may 

 exercise himself. 



