page 2 note * Traité de Fonctions Elliptiques, t. I., Chap, iii., iv. and v.

page 2 note † cf. CAYLEY'S Elliptic Functions, § 1.

page 3 note * For references to the numerous attempts to solve the problem, see Note V. in the Appendix of Müller's edition of Enneper's Elliptische Functionen (Halle, 1890).

page 7 note * As early as 1716, long before Legendre had discovered the method of multiplying and dividing elliptic functions, Fagnano was able to multiply and divide arcs of lemniscates (see Methodo per misurare la Lemniscata, pp. 343–368 of vol. 2 of his Prodnzioni Matematiche).

page 7 note † Cayley at §62 of his Elliptic Functions states the value of x and y erroneously.

page 8 note * For example, if we integrate between the limits ø=0 and then the additional algebraic function obviously vanishes, and we get where s ₁ is the fourth part of the curve.

page 8 note † Yet Legendre remarks (F. E., vol. ii., p. 591) “Le problème. … de trouver une courbe algébrique dont les arcs représentent généralement la fonction elliptique de première espèce F(k, θ) paratt n' admettre aucune avtre solution.” We shall see that we have travelled far since then.

page 9 note * The case K ² = ½ is generally solved, as we have seen, by the lemniscate which is only of the fourth degree, and its arcs express the integral F(k, φ) without any additional algebraical quantity. If we take the solution of this case given by (3) we find that the equation is

Athough this curve is not so simple as the lemniscate it has the advantage of differing little from an ellipse.

page 10 note * This is perhaps most easily seen by noticing that the condition for tangency is that (1) have equal roots, that is α⁴cos²2θ=α⁴–b ⁴, where 2θ is the angle between the tangents. Hence

2φ is equal to the angle between the tangents drawn from the centre.

page 12 note * Of course all this applies equally to the case b=∝, for then , in which case the two loops meet at the centre and we get the Iemniscate, and from (5) we see that the arc represented by σ(θ₀, θ) disappears as it ought to do since the polar equation of the lemniscate is only of the second degree in r.

page 13 note * Comparing Serret's result with Legendre's in the last section, it should be observed that if in (8) we make θ = ø, then , hence, the amplitude being a right angle, we may write s(θ)+σ(θ)=s ₁ where s ₁ is as before one-fourth of the total length of the curve. Then from (12) we get

we then as above get from (14)

From these two equations we see that, just as in Legendre's solution, by properly choosing the ends of the arc, i.e., by integrating between suitable limits, the elliptio integral of the first kind for any modulus and without any extra algebraical quantity is represented by the arc of a oassinian oval.

page 14 note * If the radius of the sphere were made infinitely great it would become a plane, and the curve on it would become a plane lemniscate.

page 14 note † This leads to exactly the same solution as Legendre's on p. 7, for let

Whence, after differentiating and reducing, we get

page 19 note * If we make n = 1 then , i.e., the lemniscate is the simplest case of this infinite class of curves. To obtain its equation we may proceed thus :– If n = l then from (13) α² + α² = 0 and αα=l, whence α² = i and α² = –i.

Putting these values in (14) we get, after one or two steps,

whence , Eliminating z between these, we get (x ²+y ²)²=C²xy.

page 21 note * We have always taken the condition ø^(m)(α)=0, but of course we take the one or the other of the conditions (11) according as m or n is the smaller. If the smaller exceeds 2 then the condition for α and α will be at least of the third degree and cannot in general be resolved, but this does not affect the reasoning by which we obtained an infinity of curves for any given value of the smaller of m or n.

Moreover, n has throughout been supposed integral, but M. Liouville proved in Liouvile's Journal, vol. x. p. 293, that n need not be integral but only rational in order that Serret's infinity of curves for x and y remain algebraic.

We have seen that ø^(m)(α)=O leads always to a relation symmetrical and homogeneous in α and α. Serret shows that this relation in its most general form is

CAYLEY (Elliptic Functions, chap, xv.) states the relation between α and α much more briefly thus:—Putting ξ for , the square of the modulus,

then the relation is

If we make m = 1 or 2, etc., then this gives the same values of the modulus as we have obtained above.

page 26 note * The best summary of Weierstrass's functions is contained in Schwarz's, H. A. Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen, nach Vorlesungen und Aufzeichnungen des Herrn K. Weierstrass (2nd edition, 1893)CrossRefGoogle Scholar. The theory of these functions is developed in Halphen's Traité des Fonctions Elliptiques. Weierstrass's Memoirs have been collected and published in Mathematische Werke von Karl Weierstrass (Berlin, Bd. I. 1894; Bd. II. 1895). In Bd. II., pp. 245–309, are two articles Zur Theorie der elliptischen Functionen. No better account of the methods of Legendre, Jacobi and Weierstrass can be found than that in Müller'a edition of Enneper's Elliptische Functionen, Theorie und Geschichte (Halle, 1890). It contains also for the student of the subject a great mass of bibliographical details. The best Memoirs in English on Weierstrass's methods are three by A. L. Daniels in vols. vi. and vii. (1884 and 1885) of The American Journal of Mathematics. Dr A. R. Forsyth has a Memoir on the same subject in vol. xxii. (1887) of The Quarterly Journal of Mathematics. In Greenhill's Elliptic Functions some parts of the modern notation are developed alongside of the old. In chap. vii. of A. C. Dixon's excellent little book on Elliptic Functions, there is a very brief sketch of the p – and ζ – functions (Halphen uses the symbol . Harkness and Morley in chap. vii. of their Treatise on the Theory of Functions devote about 60 pages to the p– , σ–, and ζ- functions.

page 27 note * Inaugural Dissertation De curvis quarum arcus integralibus ellipticis primi generis exprimuntur (Berlin 1870); Uebar Curvert deren Bogen ein elliptisches Integral erster Gattung ist (Crelle's Journal, vol. lxxix., 1875; and Berichte der naturforschenden Gesellschaft zu Freiburg, 1876).Google Scholar

page 27 note † Cf. Formeln, p. 15, or Halphen's F. E., Vol. I, p. 213.

page 31 note * cf. Halphen's F. E., Vol. I., p. 27.

From this it can be seen that since P′ω=0, P′ω′=0, P′ω″=0′ (see Formeln, p. 11) ∴ P″u, P^vu, and all the other odd derivatives of pu vanish for u=ω, ω′, or ω″But

when u=ω, ω′, or ω″.

page 32 note * We get other two sets of values that satisfy the given conditions by changing the sign of in each of the above, but of course these values would just be the conjugates of the foregoing.

page 32 note † A little consideration will show that the curves represented by this equation (which is erroneously stated by Kiepert on page 11 of his dissertation) embrace all Serrel's . For in the latter we see that there are two infinities which are the same but of opposite signs, viz., z = +α, and z=–α. So also in the curves represented by (16), remembering that , and that p′(0)=∞, we see that the infinities are , i.e., they differ only in sign.

page 36 note * Observe the analogy of this curve to the lemniscate r ²=cos2θ, which of course consists of two equal loops.

We can very easily reduce the expression for an arc u of r ³=cos3θ, to an elliptic integral of the first kind; for

Take , ∴ the above becomes

Kiepert shows in Crelle's Journal, Vol. LXXIV. (1872), how to divide an are of the curve r ³=cos3θ into 7, 13,19, 31 or, generally, 6q + l equal parts.Google Scholar

page 37 note * For the properties of minimalflächen see Todhunter's History of the Progress of the Calculus of Variations (1861); Riemann's Memoir Ueber die Fläche vom kleinsten Inhalt bie gegebener Begrenzung, revised by Hattendorff, K., Bd. 13 der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen (1867);Google Scholar and particularly Schwarz's, H. A. article Miscellen aus dem Gebiete der Minimalflächen, at pp. 168–189, vol. i.Google Scholar, of his Mathematische Abhandlungen (Berlin, 1890).Google Scholar

page 37 note † For the development of p′(u–a) see Formeln, p. 11, and for the other three terms we use the theorem that if f(u) be an elliptic function of u, then its development in the neighbourhood of u=a is

(Cf. Forsyth's Theory of Functions of a Complex Variable, p. 50).

page 40 note * Formeln, p. 10, and footnote p. 26.

page 40 note † It ought perhaps to have been mentioned that is a contraction for .

page 41 note * Formeln, p. 11.

page 41 note † Formeln, p. 10.

page 42 note * Formeln, p. 13.

page 42 note † Formeln, p. 14.

page 43 note * Formeln, p. 30.

page 44 note * Compare these values with Kiepert'a values on p. 32 for the group of curves which includes the lemnisoate.

page 46 note * Halphen'a F. E., p. 27.

page 48 note * Lilienthal does not remark that if , then the radius of this sphere is infinite and the above curve becomes a plane lemniscate, for its modulus is and the above equations for x and z (y of coarse vanishes) are the same as those for the plane lemniscate on p. 44. See also first footnote on p. 14.

page 56 note * For the development of p′(u) in the proximity of u=a, cf. footnote, p. 37, keeping in mind that P(u) has only even derivatives as already pointed out.

page 61 note * Taking for granted that v is even ; of course it does not matter whether it is odd or even—it is merely a question of which sign we shall use.