502 Notices respecting New Books. 



for the partial products of two vectors. This notation has been 

 discussed by Mr. Heaviside in the ' Electrician ' (December 9th, 

 1892), by Prof. Alfred Lodge in ' Nature' (November 3rd, 1892) 

 and elsewhere. The present paper is in part a rejoinder to the 

 statements of these gentlemeu. Then followed a paper on ' The 

 Imaginary of Algebra,' and ' The Fundamental Theorems of 

 Analysis Generalized for Space.' 



The author here proposes " to review critically the historical 

 definitions of the trigonometric terms, and the definitions, tri- 

 angular, circular, or hyperbolic, given in the best modern treatises 

 at my command ; then to devise a logical system of definitions 

 which will apply to space- analysis and that modern trigonometry, 

 which, as Prof. Greenhill shows (Diff. and Int. Calc. p. 61), 

 includes the properties both of circular and hyperbolic functions, 

 and will be able to bring within the same domain the properties 

 of the elliptic, general hyperbolic, and other functions." 



This wide extent is not gone over here, but attention is mainly 

 given to Plane Trigonometry : Trigonometry in space is handled in 

 a further pamphlet entitled ' The Principles of Elliptic and 

 Hyperbolic Analysis.' 



Dr. Macfarlane discusses in detail the treatment adopted by 

 De Morgan and more recently by Dr. Hobson, Messrs. Levett and 

 Davison, and Mr. Hayward (in his ' Vector Algebra and Trigono- 

 metry '). M. Laisant's Essai sur les fonctions hyperboliques is noted 

 with warm approval, and then Dr. Macf arlane proceeds to show how 

 the " several species of trigonometric functions — the triangular, 

 the circular, and the ex-circular " (using Mr. Hayward's nomen- 

 clature) " may be defined in harmony with one another. The 

 method adopted is afterwards shown to be applicable to the 

 logarithmic spiral, ellipse, and general hyperbola, and to a mixed 

 curve composed partly of a circle, partly of an ex-circle." In the 

 pamphlet which follows the one before us the method is said to 

 be applied to ellipsoidal and hyperboloidal trigonometry. 



To the Editors of the Philosophical Magazine. 

 Gentlemen, 



Will you kindly allow me to offer a word of explanation re- 

 specting the notice of Wiedemann and Ebert's ' Practical Physics,' 

 which appeared in the March number of the Philosophical Maga- 

 zine (p. 334). In writing the notice I had occasion to refer to the 

 treatise by Prof. E. Kohlrausch as seeming " somewhat antiquated 

 in a modern laboratory." It should be stated, in justice to Prof. 

 Kohlrausch, that this remark was intended to be applied only to 

 the earlier editions of bis work ; in a recently published German 

 edition the text has been revised and many new experiments have 

 been added. A similar revision of the English translation is greatly 

 to be desired. 



I remain, Gentlemen, 

 University College, Liverpool, Yours truly, 



, 'April 16th. James L. Howard. 



