224 Notices respecting New Books. 



needed ; and students have reason to be thankful that the task of 

 writing such a book has been undertaken by one in every way so 

 competent as Professor Cayley. His Treatise "is founded upon 

 Legendre's Traite des Fonctions Elliptiques, and upon Jacobi's 

 Fundamenta Nova and memoirs by him in Crelle's Journal;" he has 

 made " comparatively very little use of the investigations of Abel 

 or of those of later authors." A good deal has, however, been done 

 in treating "various points which require to be more fully discussed" 

 than they have been by Jacobi, and particularly the theory of the 

 Elliptic functions themselves, and not included in the Fundamenta 

 Nova the " theory of the partial differential equation satisfied by the 

 functions 0, H, and deduced therefrom the partial differential equa- 

 tions satisfied by the numerators and denominators in the theories 

 of the multiplication and transformation of the elliptic functions." 

 The Treatise is expressly designed for the use of students, and 

 great care has been taken to prevent them from being lost in the 

 wilderness of symbols to which the author introduces them. Thus 

 the first chapter is taken up with a general outline of the subject, 

 and the student is directed to peruse the chapter, not dwelling on 

 it, but returning to it as he finds occasion, the object being that he 

 may always have certain landmarks in view, and be the more able 

 to keep in mind the mutual relations of the parts of the subject. 

 With the same object, introductory articles of the nature of outlines 

 are prefixed to most of the chapters. The student is also directed 

 to confine his attention in the first instance to five specified chap- 

 ters, viz. 2, 3, 4, 12, 13. We will mention briefly the contents of 

 these chapters, both for the purpose of giving some notion of the 

 treatment which the subject receives at Professor Cayley's hands, 

 and as showing what he regards as a sort of first course of the 

 subject. 



fRdx 

 The reduction of I — -= to one of the three kinds of Elliptic In- 



tegrals is treated in chapter 1 2. That this can always be done the 

 author shows both by Legendre's method of supposing X to be 

 decomposed into two quadratic factors, and by a method of his own 

 based on a linear transformation of the undecomposed quartic func- 

 tion (X). In chapter 2 he establishes the well-known fundamental 

 relation, viz. if E(^) denotes an elliptic integral of the first kind 

 with amplitude <p, and if \p and /j. are the amplitudes of two other 

 elliptic functions of the first kind such that 



^)+F(W=FW, 

 then cos ^_ cos^cos;!/— sin0sin\//\/(l — & 2 sin 2 yu). 



Of this equation (the addition equation) seven distinct proofs are 

 given. It is plain that this equation enables us to determine the 

 amplitude of the function which is the sum (or difference) of any 

 two functions of given amplitude, and hence the amplitude of the 

 function which is twice a function of given amplitude, and then of 

 one which is n times a given function. Indeed it would be possible to 

 lay down a method of calculating the numerical value of a function 



