436 Intelligence and Miscellaneous Articles, 



Theory of Differential Equations. — Part I. Exact Equations and 

 Pfaffs Problem. By A. E. Eoksyth, Sc.D., F.R.S. (Cambridge : 

 University Press, 1890. Pp. xiv + 340.) 



The work before us is the first instalment in fulfilment of a 

 promise made by the author in his ' Treatise on Differential Equa- 

 tions ' (1885), and is written after the same thorough fashion as 

 the earlier volume. The author's aim is " to include every sub- 

 stantial contribution to the development of the particular subject 

 herein dealt with ; and the historical form, into which the treat- 

 ment has been cast, has facilitated the indication of the continuous 

 course of the development." He has drawn upon all available 

 sources, duly acknowledging his indebtedness to the several pre- 

 vious explorers in this field, and has further contributed investi- 

 gations of his own. A few illustrative examples have been made 

 up or culled from the memoirs above referred to. 



The first part consists of 13 chapters. The first two chapters 

 discuss the single exact equation and a system of exact equations. 

 Under the former head we have a full account of the methods of 

 solution given by Euler, Bertrand, Collet, JNatani, and Du Bois- 

 Reymond ; and under the latter head we have a generalization of 

 the above method given by Euler, with an examination of Natani's 

 method and of a theorem by Mayer in development of this 

 method. Erobenius's conditions for uncanonical form are also 

 discussed. The next 4 chapters are devoted to a historical 

 summary of methods of treating Pfaffs problem, Pfaff's reduction 

 as completed by Gauss and Jacobi, Grassmann's method, and 

 Natani's method. Chapter vii. contains the application of previous 

 results to partial differential equations of the first order. The 

 next four chapters are occupied, in order, with Clebsch's method, 

 tangential transformations, Lie's method, and Erobenius's method. 

 Chapter xii. gives an abstract of Darboux's method. The closing 

 chapter discusses Systems of Pfaffians, and is prefaced with a list 

 of the memoirs which form the principal sources of our information 

 on this subject. The whole treatise is a splendid piece of work, 

 but, from the nature of the subject, is " caviare to the general." 

 A full index makes consultation easy. We have detected only two 

 trivial typographical errors (on pp. 92 and 99). 



L. Intelligence and Miscellaneous Articles, 



ACTINOELECTRICAL INVESTIGATIONS. BY A. STOLTETOW. 



HPHE author sums up as follows the results of his researches, which 

 -*- have extended over almost two years, and some of which have 

 already appeared. 



1. When the rays of the voltaic arc fall on a plate charged with 

 negative electricity it is discharged ; this discharge is accompanied 

 by a marked fall of potential or not according as the discharge occurs 

 more or less rapidly. 



