Notices respecting New Books. 319 



consideration shows that extreme purity in any gas will always 

 be attained and maintained with difficulty. Even when the ne- 

 cessary work is small, as in the separation of oxygen from the 

 atmosphere, it is well to bear in mind that some work is abso- 

 lutely essential. The reversible absorption of the oxygen of air 

 may be effected by a substance like baryta ; but we must not 

 expect to recover the pure oxygen at the same temperature and 

 under a pressure equal to the total pressure at which it was 

 absorbed. Either the temperature must be raised, or the gas 

 must be exhausted at a pressure less than that under which it 

 existed in the mixture during the absorption. It is just possible 

 that this point might be found to be of practical importance in 

 the solution of the problem of extracting oxygen from the air. 



XXXV. Notices respecting New Books. 



An Elementary Treatise on the Integral Calculus, containing Applica- 

 tions to Plane Curves and Surfaces, with nwmerous Examples . By 

 Benjamin Williamson, A.M., Fellow and Tutor, Trinity College, 

 Dublin. London: Longmans, Green, and Co. 1875 (crown 8vo, 

 pp. 267). 



VVfE have already noticed Mr. Williamson's 'Elementary Treatise 

 on the Differential Calculus ' (vol. xliii. p. 307), and have now 

 to perform the same office for his companion volume on the Integral 

 Calculus, — a work characterized by the same excellences as those 

 which marked the previous volume. It is written with clearness 

 and accuracy, and is illustrated with an abundance of examples. Its 

 contents may be briefly described as consisting of three parts. In 

 the first are given the ordinary methods of integration, viz. by re- 

 duction to known forms, of rational fractions, by successive reduc- 

 tion, and by rationalization. In this part of the work, which 

 occupies the first five chapters, integration is treated simply as the 

 inverse of the process of differentiation. In the next part, the 

 sixth chapter, a sufficient account is given of Definite Integrals ; and 

 here the student is introduced to the notion of integration as a 

 process of summation. In the latter part of this chapter some ac- 

 count is given of the Eulerian Integrals, particularly of the second ; 

 and at the end there is a table of values of the gamma function 

 (log T (j>) + 10) from p== 1-000 to j p=l*999 to six places of deci- 

 mals ; but the method of constructing this table is omitted as " too 

 complicated for insertion in an elementary treatise." The third 

 part of the book, comprised in chapters seventh, eighth, and ninth, 

 explains the application of the Integral Calculus to finding areas and 

 lengths of curves and volumes of solids. In this part will be found 

 many useful and interesting properties of curves ; e. g. in addition 

 to those which we might expect to find almost as a matter of 

 course, we may mention Lambert's theorem on the area of an 



