Notices respecting New Books. 227 



- } would be the complete function defined from the equation 

 ar__ f * doc 



Moreover the circular functions are periodic, having for their com- 

 mon period four times this quantity, =2?r; viz. we have 



sin , . >. sin 



(w + 2tt) = u* 

 cos v J cos 



Corresponding to ^ we have in the elliptic functions in the first 



instance the complete function K ; viz. K is a real positive quantity 

 defined by the equation 



rr C 2 d <P 



Jo v o-- k2 sin2 ?y 



where K is of course not a mere numerical transcendent, but a 

 function of Jc : K is such that we have sn K= 1, en K=0, dn K=&' 



[i. e. hJ(l — Tc l ~\ , and it ultimately appears that the sn, en, 



and dn of it + 4K are the same as the sn, en, and dn of u respec- 

 tively ; or the functions have a real period 4K" (pp. 10, 11, 12). 

 Our limits do not allow of our extracting the whole article ; but we 

 may just add that, as well as the analogies between the circular 

 and elliptic functions, attention is drawn to the transformation 

 theory, and the second period 4(K+ V — 1 .K') in the elliptic func- 

 tions, which have no analogues in the circular theory. 



An Elementary Treatise on the Differential Calculus, containing the 

 Theory of Plane Curves, with numerous examples. By Benjamin 

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

 Third Edition, revised and enlarged. London : Longmans, Green, 

 and Co. 1877. (Crown 8vo. Pp. 416.) 



We have already noticed the first and second editions of this ex- 

 cellent work, and have now to notice the third. As might be 

 expected, the work in its present form is not substantially different 

 from what it was when it first appeared. In fact, a considerable 

 part of the second and third editions agree page for page ; so that 

 the amount of difference may be pretty accurately inferred from 

 the fact that the former edition contains 367, the latter 416 pages. 

 The alterations are made in two ways -.—first, by small additions 

 and occasional omissions here and there ; secondly, by the insertion 

 of a few articles of considerable length. Of the first kind w 7 e may 

 notice such changes as these : — the insertion of the proof of a con- 

 dition that Vdoo + Qdy may be an exact differential (p. 143); the 

 recasting and extension of the article on the curve r m =za m cos mQ 

 (p. 227)"; the additional examples of algebraical maxima and minima 

 (p. 163) ; the additions to and omissions from the examples on 

 Partial Differential Coefficients (pp. 137-40), and so on. Of the 

 second kind may be mentioned : — two long extracts from M. 

 Navier's Lecons cV Analyse on the principles of the Differential 

 Calculus (pp. 7-10), and on the Failure of Taylor's Theorem 



Q2 



