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XXX. Notices respecting New Books. 



An Elementary Treatise on Elliptic Functions. By Akthur Cat- 

 ley, Sadlerian Professor of Pare Mathematics in the University of 

 Cambridge. Cambridge : Deighton, Bell, and Co. London : 

 Bell and Sons. 1876. 8yo. Pp. 384. 

 TX most works on the Calculus the subject of Elliptic Integrals is 

 -*- either altogether excluded or treated inadequately; and not 

 merely is this the case with works that are in fact elementary, but 

 even in the elaborate treatise of the late Professor De Morgan the 

 subject is dismissed in two paragraphs, which are in substauce as 

 follows : — " Important as Elliptic Integrals are in certain classes 

 of problems, and numerous as have been the properties of them, 

 which have been investigated, it cannot yet be said that either these 

 problems or methods lie so close to the grand route on which the 

 students' elementary course should be marked out as to require a 

 detailed treatise on them to be inserted here." He then goes on 

 to state : — that an Integral is called Elliptic when it can be put into 



JRclv 

 —Tsr, where E is a rational function of x, and X a 



rational and integral function of the fourth degree ; that it is capable 

 of being shown that the actual calculation of all such Integrals is 

 attainable when Tables of the following integrals (called elliptic 

 integrals of the first, second, and third kind respectively) have been 

 constructed, viz. 



i 



«P 1 d<f> 



1 + n sin 2 (p ^(1—Jc 2 sin 2 <f) 



in which h (the modulus) is less than unity, and (the amplitude) 

 does not exceed \ * ; and that extensive Tables of the first two kinds 

 have been given by Legendre, with methods of approximating to 

 integrals of the third kind (pp. Q5Q, 657). In fact Legendre worked 

 at the subject, systematizing and supplementing the work of his 

 predecessors and making the actual numerical calculations, for 

 about forty years. His results, in their final form, were published in 

 1825-26. They were scarcely out when the subject was treated from 

 an entirely new point of view by Jacobi, whose Fandamenta Nova 

 was published in 1829, being preceded and followed by memoirs 

 from 1828 to 1858, and by Abel, whose memoirs appeared from 1826 

 to 1829. 



It will be seen from this statement that the student who wishes 

 to make out the subject will not gain much help from the ordinary 

 textbooks ; and indeed not much has been written (we believe) on 

 the subject in English, beyond the works referred to in the note to 

 the above-quoted passage in De Morgan's ' Differential Calculus.' 

 An Elementary Treatise on Elliptic Integrals was therefore much 



