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XLIII. Notices respecting New Books. 



An Elementary Treatise on the Integral Calculus, containing Appli- 

 cations to Plane Curves and Surfaces, with numerous examples. By 

 B. Williamson, E.B.S. Third Edition, revised and enlarged. 

 London : Longmans, 1880. 



WE used at Cambridge, "in the consulship o£ Plancus," to 

 divide " articulate speaking " men into two classes — those 

 who understood the Differential Calculus, and those who did uot. 

 Such a division still holds good ; and possibly the relative magnitudes 

 of the two divisions are still the same, though Todhunter has, since 

 the. epoch to which we refer, brought out his valuable books on 

 the two calculi. An English student has still no great choice if he 

 confines his reading to works in his own language. De Morgan's 

 treatise remains still a vast repertory which repays occasional ex- 

 ploration; but who could take it as an ordinary text-book? Tod- 

 hunter's works referred to above present the subject in a very 

 clear light to students, and are enriched by a choice collection of 

 exercises. Putting the four American text-books we have at hand 

 (which are mainly founded on the English text-books and Bertrand's 

 classical work) on one side, the only serious rival to Todhunter is 

 the work before us. That its merits are of a very high order, and 

 have been recognized by teachers and students, is evidenced by its 

 having so soon reached a third edition. Its bibliography is as 

 follows : in 1875 appeared a compact little work of 267 pages in 

 nine chapters ; in 1877 the work had grown to 348 pages, had 

 developed a preface, and had increased to eleven chapters, the two 

 new chapters treating of Moments of Inertia (a new subject in 

 elementary treatises on the calculus) and Mean Yalue and Pro- 

 bability. For the former of these additions the author was mainly 

 indebted to Prof. Townsend's papers in the ' Quarterly Journal of 

 Mathematics ;' and the latter was written by Prof. Crofton, whose 

 exceedingly ingenious papers in the Phil. Trans, especially fitted 

 him for this contribution. The third edition preserves the same 

 order and number of chapters, but has incorporated numerous im- 

 portant additions, reaching to 371 pages, with a useful index at 

 the end. These additions are all improvements ; and so the work 

 vires acquirit eunclo. "We shall not enter in any detail upon the 

 contents and method of a work which has so rapidly risen into 

 favour, but shall content ourselves with merely indicating one or two 

 of the more important novelties. We commence with Frullani's 

 theorem in Chapter vi. (on Definite Integrals): here Mr. Williamson 

 refers to Mr. E. B. Elliott's contributions on the subject to the 

 * Educational Times,' and his extensions of formulae, given in this 

 chapter, to multiple definite integrals (Nos. 106 and 113 of the Pro- 

 ceedings of the London Mathematical Society), and to Mr. Leudes- 

 dorf's paper on the Extension of Frullani's Theorem (No. 131, Pro- 

 ceedings of the same Society). Some of these results have been 

 inserted in the text. In the same chapter is an expression for the 



