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10. The formulæ, which have just been obtained, yield the following numerical results in the special cases in which k has its largest possible value, namely or (n-1); the terms containing negative powers of n are all those which need be retained in calculating the expressions to six places of decimals when n≥ 10:

(I.) When is even and kn, then*

Hn, in= 0.18323, 39 × n log1n + 0.06515, 65 × n

Mn, bn=0,


+0.02181, 66 × n ́1 — 0.01758, 46 × n3+.....,

Nn, in 0.03392, 22 x n3-0.01326, 29 x n

+ 0.01527, 16 × n ̄1— 0.03708, 31 × n3+....

(II.) When n is odd and k = (n − 1), then Hn, (n-1)=0.18323, 39 x n log1, n+0.06515, 65 × n -0.07635, 82 x n +0.02278, 82 × n3 +...,



Mn, (n-1)=0.05515, 89 x n 0.06544, 98 × n ̃1

+0.021, 5321 x n3+...,

Nn, ¿(n−1) = 0.03392, 22 × n3-0.06842, 18 x n

+ 0.03163, 41 × n ̃1+0.022, 1303 × n2+....

In case the expressions are ever required to a high degree of accuracy, we give the coefficients in their large terms correct to 20 decimal places:

(III.) When kan or (n-1),


Hn, k = (4π ̄1 log ̧10) ×n log1n+ñ ̈1(log ̧4−logπ+y) × n + .....


=0.18323, 38997, 19856, 93522 × n log1n

+0.06515, 64533, 06926, 22763 × n + ... .

(IV.) When k = 1 (n − 1),

Mn, (n-1)=(log ̧2) ×n +...

= = 0.05515, 89000, 38162, 89835 × n + ....

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The method of evaluating the coefficient of n3 in N,in is given below in

§11; it is there shewn that its value is 70,/ (8π3).

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-0.06842, 18119, 62487, 50966 × n +....

11. It is worth noting that a simple method of evaluating the coefficient of n3 in Nn, n is to use the Fourier expansion

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and integrate term-by-term (this procedure may be justified by the device employed by Whittaker and Watson, Modern Analysis, p. 163, sinall type), so that we have

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which is the result stated. The value of σ, has been given by Stieltjes, Acta Mathematica, X., to 32 places of decimals.




IE major part of this paper is devoted to an extension to residues of double integrals belonging to an algebraic surface, of a well-known theorem, according to which the sum of the logarithmic periods of an Abelian integral is equal to This extension forming the object of Part I. is embodied in Theorem II. In the simplest case it has already been given by Poincaré and Picard, though not explicitly stated by either. In Part II. the converse proposition is taken up,


* Picard et Simart. "Traité des fonctions algébriques de deux variables," vol. i., pp. 52-55. Further references will be found there.

and a short discussion on cyclic residues is given. It may be said that the need of this investigation arose in connection with a study of the properties of triple integrals belonging to an algebraic variety.

1.-Point residues, Fundamental theorem.

1. Let F(x, y, z)=0 be the equation of an irreducible algebraic surface F, and C,, C,, ..., C, irreducible curves of F. The curve C, will have for projection in the xy plane a curve of which the equation is g(x, y) = 0, and there will be in general another curve C, of F having the same projection. Any double integral of algebraic function J belonging to F, infinite on some of the C's, can always be put in the form

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where P(x, y, z) is a polynomial adjoint to F such that the double integral is finite on the curves C. The residues* of J with respect to C, are the periods of a certain Abelian integral relative to C. The logarithmic periods of the latter are the polar residues of J with respect to either the multiple points of C or its intersections with the other C's. The cyclic periods of the Abelian integral are the cyclic residues of J with respect to C. From the proposition already referred to, follows:

THEOREM I. The polar residues of a double integral belonging to the surface, relatively to points on a given irreducible curve, have a zero sum.

2. Let O be a multiple point of the composite curve, formed by the aggregate of the C's. Such a point, together with the branches of the C's through it, will be called a star, the branches of the C's through O being the branches of the star. The Abelian integral already considered may have several logarithmic periods with respect to 0; at least if this point is multiple for C, and in that case J will have several residues with respect to the star, each corresponding to one of its branches. The fundamental theorem may be stated thus:

THEOREM II. The residues of a double integral with respect to an ordinary point of an algebraic surface have a zero sum.

3. Consider first the case of a two-branched star of vertex 0, formed either by a double point of one of the curves, say C1,

* For their definition see Picard-Simart, loc. cit., also vol. ii., p. 203.

or by the intersection of two of them, say C,, C. We have identically

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The residues of the second integral are all zero.* Hence those of J and of the first are the same. By a repeated application of this identity we can replace J, as far as its residues are concerned, by a double integral of one of the two following types:

R(x, y, z) dx dy

[R (x, y, z) dx dy, JR (x,


g1(x, y)

g1(x, y)g, (x, y)

where R is in both cases a rational function finite in the

vicinity of O. For each of these the proposition has already been proved by Picard,† and it may be verified that his proof holds whatever the singularities of F, provided O is an ordinary point of the surface.

4. Before we proceed with the general case a remark will be necessary. Let C be a twisted curve with arbitrary singularities in ordinary space, O one of its multiple points. Consider a quadratic transformation defined as follows:Given a point M of space, its transformed M' is obtained by joining M to O, and taking the intersection of OM with the polar plane of M with respect to a fixed quadric Σ. We can reason with transformations of this nature and the given curve, as usually done in proving Nöther's theorem on the reduction of singularities of plane algebraic curves, and show that C can be transformed into a curve C', such that the vicinity of O on C be transformed into the vicinity of a finite number of ordinary points on C'.

5. To prove our theorem in the general case we propose to reduce it to the case considered in § 3 by means of transformations such as the preceding.

Let O be an arbitrary star vertex, and consider the quadratic transformation of which O and an arbitrary quadric Σ

* Picard-Simart, II., p. 204.

+ Picard-Simart, I, pp. 54-57. Thus, for the second, the proof is obtained by interchanging P and Q in the expression at the bottom of p. 55, for then its sign changes.

See, for example, F. Severi, Lezzioni di geometria algebrica; Padova, Angelo Draghi, p. 60. Instead of the curve differing very little from the given one, considered by Severi, p. 61, it is sufficient to take a cone differing very little from one projecting the curve C. Otherwise little is changed.

are the fundamental elements. Let us consider the effect of the transformation, which we denote by T, upon the elements involved in the theorem.

(a) Effect of Tupon F and J. Let be the polar plane of O with respect to 2, t the plane tangent to F in 0. The surface FT(F) has for multiple curve the conic (π, 2), and for simple line the line D=(t,). The double integral J becomes a double integral J' relative to F", with certain stars and residues to be determined.

(b) Effect of T upon the stars. Let CT(C). The reducible curve formed by the aggregate of the C's on F" has a certain number of multiple points, and the branches of the C's through them form the stars of J'—stars which are clearly the transformed of those on F. If on the latter a star vertex other than O is not on the conic (π, Σ), which can be assumed for all stars, its vertex and branches are transformed into the vertex and branches of a star of F'.

Among the stars of J'a certain number 0,', O2', ..., Ο'. have their vertex on the line D. The branches of the C's, together with D, form the stars on F transformed of the star 0. None of the stars O' has branches other than D, tangent to D, if the branches of the star O have all their "cycles" of order unity.*

(c) Effect of T upon the residues of J. A two-dimensional cycle I, corresponding to a residue of J with respect to C1, is generated in the following most general way*:-Let σ be a closed circuit on the Riemann image of C,, M any point of o. Through this point pass an element of area contained in the four-dimensional real image of F, and on this element draw a very small closed circuit, surrounding the point M. The locus of , when M describes σ, is the superficial cycle г. This cycle will correspond to a point residue of J if o is entirely situated in the vicinity of a point of C1, or more precisely if is reducible, by steady deformation, to a point of C, It is clear that the effect of 7' is to transform a closed circuit o' of C', M into a point M' of o', into a small closed curve on an element of area through M',


*The coordinates of a point in the neighbourhood of (x, y, z) on the twisted curve Care gived by developments of the type

x− x。=alλ+a ̧tλ+1+..., y—yo=b_t"+..., z−2。=coto +.....

in positive integral powers of a parameter t. The smallest of the numbers A, u, v is called the order of the cycle defined by these developments. See E. Picard, Traité d'Analyse, vol. ii., p. 358, first edition. Also F. Severi, loc. cit, p. 70. Plane curves alone are considered there, but the extension of these notions, due to Halphen, is obvious.

f Picard-Simart, I., pp. 58-60.

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