S41 



ELLIPTIC FUNCTIONS. 



ELLIPTIC POLARISATION. 



84S 



kind out of a large number which have .been proposed. The first la 

 the simple and rough method suggested by the first property iu the 

 article ELLIPSE. Let two pins be fastened to the paper at the points 

 in which the foci are to lie, and let a thread, equal in length to the 

 proposed major axis, have one end tied to each pin. Then if a pencil 

 move in such a way as to keep the thread always stretched, it will 

 describe an ellipse. 



The second method is as follows : it is known that if any two fixed 

 points in a straight line A be made to move along two other straight 

 lines, B and c, then every other point in A will describe an ellipse. If 

 then two grooves be made (at right angles to each other, for con- 

 venience), and if two pins attached to a ruler be made to travel in the 

 grooves, the motion of the ruler will make any pencil attached to it 





trace out an ellipse. The distances of the pencil from the two pins 

 will be the semiaxes of the ellipse. If the pins be attached to the 

 ruler by clamping screws their distance may be altered, and the instru- 

 ment may be made to describe any ellipse within limits depending on 

 the length of the ruler and of the grooves. 



ELLIPTIC FUNCTIONS, or ELLIPTIC TRANSCENDANTS. 

 This is no subject for a Cyclopaedia, except in a very limited sense. 

 We can only undertake here to describe the general character of the 

 subject. 



If we had passed direct from algebra to the differential calculus, 

 without any consideration of logarithmic or trigonometrical quantity 

 [TBAXSCEXDEJJTAL], we should have found ourselves stopped in the 

 integral calculus by want of language in which to express the integral 

 of such a function as 



dx 

 V (a + bx + C3?). 



It would no doubt have been found (since the same difficulty has 

 been conquered in a more complicated form in the subject of the 

 present article) that all such integrals could be expressed by means of 

 those of 



dx dx 



V (a* - 



and 



and the connection of the former of these with the arc of a circle, and 

 of the latter with logarithms, the properties of the functions called 

 nines and cosines, tables of their values, with a complete system of 

 trigonometrical formulae, would speedily have followed. From this 

 mode of arriving at a new set of transcendentals we are saved by 

 having the science of trigonometry already prepared on a geometrical 

 basis, in connection with the properties of the circle. But we have no 

 such preparation founded on the ellipse, a curve which is an extension 

 of the circle. Nor is there any connection between the arc of an 

 ellipse and the excentric anomaly, the angle of which is the extended 

 representative of the angle at the centre in the circle, except by means 

 of a definite integral. In fact, a being the semi-axis major, e the 

 excentricity, <f> the excentric anomaly, and s the arc, we have 



t=af / (1 e* cos 2 <t>) . dip. 



The integral calculus is, as yet at least, the only manner in which 

 the arc of an ellipse can be approached : accordingly, a large class of 

 integral/I, closely related to, and containing among them, the expression 

 for the arc of an ellipse, have received the name of elliptic functions. 



By an elliptic function is meant any integral of the form 



P dx 



in which p a a rational function of .<-. This can be shown to depend 

 for its determination upon the form 



V (I c* sin s <t>)' 



in which Q is a rational function of sin- <p, and e is less than unity- 

 This, again, can be shown to depend upon 



'A + B sin 2 <f> d<f> 



i + n sin 2 (f> ' V(l e 2 sin 2 0,)' 



which in every case depends upon one or more of the three following 

 forms : 



dip 

 V (1 c 3 sin 2 $>) J V (1 e 2 sin 2 <p) . d<f> 



1 dtb 



1 + ldn 2 </> ' / (1 e 2 sin 2 <<>)' 



which are called functions of the first, second, and third species. 

 Every elliptic function represents the arc of some algebraical curve. 



The earliest researches into the integrals connected with the arcs of 

 an ellipse or hyperbola are those of Maclaurin, in his treatise on 

 Fluxions, and of D'Alembert, in the Berlin Memoirs for 1746. 

 Fagnani, in 1750, showed how two arcs of an ellipse might be assigned 

 in an infinite number of ways, which should have for their difference 

 an algebraical expression. Euler, in 1761, showed how to assign the 

 complete integral of certain differential equations of which the terms 

 are separately nothing but elliptic functions. Landen, in 1755, showed 

 that every arc of an hyperbola can be obtained by means of two arcs of 

 an ellipse. Lagrange, in 1785, gave a general method for approxi- 

 mating to the values of elliptic functions of all kinds. 



But those to whom it is due that the theory of elliptic functions has 

 become a distinct and important branch of the integral calculus, with 

 general formulae which have made it an extended form of trigonometry, 

 are Legendre, Abel, and Jacobi. Of these, the first almost devoted his 

 life to the subject. His various memoirs, and his latest extensions of 

 them, are contained in the two following works : 



' Trait(5 des Fonctions elliptiques et des Integrates Euldriennes,' 4to, 

 Paris, vol. i., 1825; vol. ii., 1826; vol. iii., containing three supple- 

 ments, 1828; and ' Exercices du Calcul Integral,' 3 vols. 4to, Paris, 

 1811. Both these works contain extensive tables for the calculation of 

 the functions. 



The memoirs of Abel, the substance of some of which are in the 

 third volume of Legendre, were originally published in Crelle's journal, 

 and are now collected in his works, which were published in French, 

 ' O2uvres de N. H. Abel,' collected by B. Holmboe, Christiania, 2 vols. 

 4to, 1839. Jacobi's work is ' Fundament* nova theoriae functionum 

 ellipticarum, auctore D. C. G. I. Jacobi,' Regiomonti, 1829, 4to. As 

 to elementary works, there is some account of Legendre's first memoirs 

 in the second and third volumes of Lej bourn's ' Mathematical Reposi- 

 tory : ' several works on the integral calculus, that of Mr. Hymers in 

 particular, contain the first elements. There is a work on the subject 

 expressly by M. Verhulst, printed in Belgium ; and there is an article 

 on elliptic functions and definite integrals in the ' Encyclopaedia 

 Metropolitana.' 



Since the publication of the ' Penny Cyclopaedia ' many researches 

 have been made, which not only illustrate the analogy between elliptic 

 functions and ordinary trigonometry, but advance into higher views of 

 transcendental integration, and foreshadow the time when elliptic 

 functions themselves will be but a particular case of a higher theory. 

 There is an able report on the progress of this branch of analysis up to 

 near 1846, by the late Robert Leslie Ellis, in the Reports of the British 

 Association for 1846. There is also an elementary work, or at least as 

 nearly elementary as its mode of view will permit, by MM. Briot and 

 Bouquet, ' The'orie des Fonctions Doublement pcriodiques, et, in par- 

 ticulier, des fonctions elliptiques,' Paris, 1859, 8vo. 



ELLIPTIC POLARISATION is the name given to a peculiar modi- 

 fication of polarised light, which is thus designated because, according 

 to the undulatory theory, the ethereal particles describe ellipses in the 

 case of light so modified. The term was, however, employed by Sir 

 David Brewster for another reason ('Phil. Trans.,' 1830); and ellipti- 

 cally polarised light is recognisable as such by experimental characters 

 independently of any theory as to its nature. 



If ordinary polarised light, which, in contradistinction to light that is 

 elliptically polarised, is called plane-polarised, is reflected from the sur- 

 face of glass or any other transparent medium, the reflected light 

 (neglecting for the moment certain small deviations to be noticed 

 presently) is plane-polarised, having all the characters of light polarised 

 by reflexion from glass at the proper angle, or by transmission through 

 a Nicol's prism, &c. If, however, plane-polarised light be similarly 

 reflected from a metal, the reflected light will no longer be found to be 

 plane-polarised, unless the plane of incidence be parallel or perpendicu- 

 lar to the plane of primitive polarisation, or the angle of incidence be 

 very small or very near 90". In general, on being examined by an 

 analyser, such as a Nicol's prism, which is made to rotate, the light is 

 never extinguished, but merely passes through a minimum, being a 

 maximum and a minimum alternately at every quarter revolution. So 

 far it resembles partially polarised light, such as common light reflected 

 from glass at an angle differing from the polarising angle, or reflected 

 from a metal. But the two kinds are in reality totally different, as 

 may be seen at once on examining them by a Nicol's prism capped 

 with a plate of calcareous spar, cut perpendicularly to the axis, which is 

 turned towards the light to be examined. Partially polarised light (or, 



