;6o 



NA TURE 



[February 15, 1894 



of analysis as are susceptible of application to physics, 

 easily intelligible to students of that subject. The 

 present book, addressed, we are told, to the trained mathe- 

 matical student, is stated to be primarily a collection of 

 problems (mostly in dynamics and electric flow) whose 

 solutions are expressible by elliptic functions ; and it is 

 intended that the properties of these functions should be 

 suggested by, and developed simultaneously with, the 

 problems in hand. Really, of course, it is much 

 more. In fact, the student who works completely 

 through the book will meet with a good many of 

 the formulae of common occurrence in the ele- 

 mentary part of the subject, and will, moreover, learn 

 to manipulate them for himself ; and whether he be 

 interested most in the motion of tops, or the stability of 

 ships, or the biquadratic form, he will probably be sur- 

 prised at the amount of information condensed here. 



The book opens with a consideration of the motion of 

 the common pendulum. The fact that in this motion 

 the angular displacement depends uniquely upon the 

 time, suggests the inversion of the elliptic integral ; the 

 existence of a real period of the functions thus obtained, 

 is suggested by the periodic motion of the pendulum. 

 The functions are then immediately used to express the 

 solution of Euler's equations for a body moving about a 

 fixed point under no forces. Then follow seventy pages 

 devoted to the expression of elliptic integrals in terms of 

 the functions, in the course of which, beside a vast 

 variety of examples collected from Legendre and else- 

 where, are found a consideration of Watts' Governor, of 

 the Elastica, of the Sumner lines on a Mercator chart, of 

 the Catenoid, of quadrantal oscillations, and of other 

 things — the notation being sometimes Jacobian and some- 

 times Weierstrassian. It is needless to say that here is a 

 mine of wealth for the examiner. It is only in chapter 

 iv., when we are a third of the way through the book, 

 that the addition theorem of the functions becomes 

 necessary. And while this is proved by a pendulum 

 view of Jacobi's two-circle method, space is found for a 

 thorough examination of Legendre's method and a de- 

 tailed account of the porismof thein- andcircum-scribed 

 polygon for two circles, the diagrams being of the most 

 painstaking character. Then follow sixty pages which 

 will be perhaps the least interesting of the book — at least 

 to the students for whom Prof. Greenhill writes— devoted 

 to an algebraic exposition of the addition theorems for 

 the three kinds of integrals. They contain an examina- 

 tion of the theorems of Fagnano and Graves for the 

 ellipse and hyperbola. They are followed by an account 

 of the tortuous elastica, succeeded by a resumption of 

 the motion of a body about a fixed point under no forces, 

 wherein the author introduces a very full account of the 

 herpolhode. In the hundred pages remaining, the book 

 may be said to be drawing to a conclusion, the double 

 periodicity is considered, Cartesian ovals being intro- 

 duced in connection with the expression of functions of 

 a purely imaginary argument ; a chapter is devoted to 

 the factor expressions of the functions, here suggested 

 by hydrodynamical considerations ; and the last chapter 

 is a summary of the earlier part of the theory of trans- 

 formation, characterised, however, like the rest of the 

 book, by the utmost particularity, numerical and other- 

 wise. 



NO. 1 268, VOL 49] 



This summary will show to some extent the scope of 

 the work. It is essentially a student's book, written in a 

 concise conversational style ; but whether the student 

 have more sympathy with physical or pure mathematics, 

 he cannot fail to find much that is new to him, and be 

 surprised at the detail with which it is given ; and the air 

 of practical reality which pervades every page, and the 

 skill and originality with which the results are obtained^ 

 will atone for the tentative nature of many of the de- 

 monstrations. 



It is, in fact, in this regard that the reader may be most 

 unfair to the author. It is no part of his plan to develop- 

 any demonstration beyond the nearest point at which it 

 suggests the formula required, or to use any more general 

 method of enquiry than is absolutely necessary, or to re- 

 gard the subject in any other way than as a collection of 

 formulae. To forget this is to wish for many things to be 

 ditferently stated — is, indeed, to wish for a quite different 

 book. A few instances will suffice. The author frequently 

 makes the remark that the present state of the theory is 

 due to Abel's brilliant idea in inverting the elliptic integral 

 of the first kind. One fears that the reader may enquire 

 whether the inverse function is a one-valued function of 

 its argument for all the values of the latter, or may forget 

 that the expansion of p. 202 is not valid for all values of 

 the quantities involved. He may even wish to invert the 

 integral of the second kind, notwithstanding that it is 

 here expressed in terms of the integral of the first kind. 

 Or, again, the statement on p. 266, that and \|/- " satisfy 

 the conditions required of the potential and stream func- 

 tions," may lead to misconception, for it is not sufficient 

 that <^ be infinite at A and C ; it must be infinite in the 

 neighbourhood of A like a multiple of the logarithm of 

 the distance from A. And in the same way, on p. 281, in 

 attempting to realise how a " uniform streaming motion 

 parallel to the vector ina " is consistent with the motion in 

 the strips which is represented by the other factors, we 

 are liable to desire a proof that functions whose equality is 

 not identical, but, as here, the result of proceeding to a 

 limit, necessarily represent the same fluid motion. The 

 fact is that the two functions considered here are not 

 equalfor2' = oo , where sine^-has amost essential singularity. 

 Or, again, we may wish that the signs had received more 

 attention ; as, for instance, on p. 24, or throughout 

 chapter ii., and in many other places. And this the more 

 that Jacobi himself is known to have printed a mistaken 

 sign (for en (K + iK') ). And this wish is not allayed by 

 the fact that in the reservation of these difficulties, made 

 on page 45, poles and branch points are mentioned to- 

 gether, as if similar singularities. While, lastly, if we 

 forget the object of the book, we shall most devoutly wish 

 a better recognition of the fact that the Jacobian 

 functions and the Weierstrassian functions are not the 

 fundjfmental fact of the theory. Underlying both is the 

 same algebraic irrationality, now expressed by a binodal 

 quartic, and now by a cubic curve — from either of which i 

 both these functions and many others can be con- ! 

 structed, the distinguishing mark being only the number 

 and position of the poles. One does wish indeed that ' 

 Prof Greenhill had found occasion to state somewhere ; 

 that the algebraic method he adopts throughout, fascinat- 

 ing as it certainly is, is also, in the strict sense employed 

 by him, of only antiquarian interest, in view of the de- ; 



