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The condition that the line v const. is tangent to an asymptotic curve of the given surface at (0, μ) is that v satisfies the equation*

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i=1, 2, 3.

0 ̧ = 8b ̧ μv+c; +2d ̧μ+e̟¡μ3

On expanding the above determinant we obtain

2μv3+4v [b2u*—b ̧μ3 + (b,—a ̧) μ3+a‚μ-a,]

-[be̟‚μ3 — (be̟ +2е ̧) μ3 + (b ̧¢ ̧+a ̧e̟ ̧+b ̧ e ̧ + 2e,—4d ̧) μʻ


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If we substitute in (11) the value of v from (12) we obtain the required equations of the surface of asymptotic tangents along the conic λ=0. The sextic plane curve (12) in current coordinates (u, v) has a fourfold point and a consecutive double point at vco. It has no other multiple points and is of genus three. The generators of the surface of asymptotic tangents are in (1, 1) correspondence with the points of the curve (12), so that the ruled surface is of genus three.

To the sections by two given planes Σax=0 and 2ẞx.=0 of the ruled surface defined by (11) and (12) correspond the

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where is defined in terms of μ by (12). If we eliminate σ and between these equations and (12) we obtain au equation of degree ten in μ, each of whose roots determine an intersection of the line Zax=0, 6x=0, with the surface of asymptotic tangents. This surface is thus of order ten.

*Cf. Wilczynski, "Projective differential geometry of curved surfaces." Transactions of American Math. Soc., vol. viii., pp. 233-210 (1997).


By C. E. WEATHERBURN, Ormond College, Melbourne.


A PREVIOUS paper* of the author's brings out the close

analogy that exists between the behaviours of the

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This intimate connection suggests the enquiry for a series of functions of the radius vector r from a fixed point, bearing the same relation to the former that the integral powers of r bear to A'. In the first part of this paper it will be shown


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C being a constant depending on m; so that the series of functions

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is the desired series such that the mth satisfies the equation.

Dim u = 0,

except at the pole from which r is measured.

Observing that the ordinary corollary to Green's theorem can be equally well written.

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we shall obtain in the second part of the paper a set of extensions of Green's theorem involving the operators Dm,

"Green's functions for the equation Au-k?u=0, and the integration of some related equations," Quar. Jour. of Math., vol. xlvi., pp. 198-215.

and expressing the value of a regular function u (p) in terms of Du and the boundary values of

d dn

D-u and Diu, (i=1, 2, ..., m),

at the same time establishing the unicity of the solution of D1m u = ƒ (p),

given the boundary values of u and its first (m-1) normal derivatives. An extension of Gauss' surface integral theorem is also obtained.

In the third part the Green's function, G, of order m for the equation




is defined. The definition is made with reference to the functione mentioned above; and a general formula is proved giving the value of a function u, regular along with all its derivatives of order up to (2m-1), in terms of G and the boundary values of certain derivatives of u. Properties of the Green's function G, are established agreeing with those of part two of the paper referred to.


These results are in the fourth part applied to the integration of certain equations of order 2m, the solution being effected in general by means of the function G There are some important points of difference according as m is even or odd.


Certain functions analogous to harmonic functions.

§1. I propose first to examine a class of functions bearing the same relation to the operator D' A'-k' that harmonic functions bear to A'. A function which in a given region is regular along with its first derivatives, and moreover satisfies the equation

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will be called "symphonic" in that region. If it is regular along with its derivatives of all orders up to (2m – 1) and also satisfies the equation

DTMm u = 0,

it will be called poly-symphonic of order m, or briefly m-symphonic.

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Let be the distance measured from a fixed point q to a variable point p, and suppose that q is outside the region



considered. Then it is easily verified that is symphonic.


Further, e is bi-symphonic, as can be shown by using the formula

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u being a function of only, and the dashes denoting successive differentiations with respect to r. We thus find

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and since the second member is symphonic it follows that e is bi-symphonic. By the same procedure it may be shown. that reis tri-symphonic.

More generally we may establish by induction that (mer) is m-symphonic. For in virtue of (2) it follows


(3) (A'-k') (2-3 e ̄kr) = -2k (m −1) (2-3 e-kr)


.m-2 kr


+ (m − 1 ) (m − 2) μm−♦ ̧ ̄kr, from which we conclude that if 7-3-kr is (m-1)-symphonic and me is (m-2)-symphonic, then e is necessarily m-symphonic. But rer is tri-symphonic and e -kr bi-symphonic, so that the result is established. We may also note that the expression

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where the c's are arbitrary constants, is m-symphonic except at the pole.

Operating again on (3) with D' we find that



D* (¿ ̄3 e ̃kr ) = ( — 2k)2 (m — 1) (m − 2) μm→± e ̄kr + Qm-39


Do ( 2•TM−2 e ̄ kr) = (−2k)3 (m − 1 ) (m − 2) (m −3) (μm−3¿ ̄kr) + Qm-4?

where Q, consists of terms poly-symphonic of order n at most; and finally

D ́m−3 (pm−3 ¿ ̄kr) = ( — 2k;)m-1 (m − 1 ) ! —


=(-2k)"12π (m−1)! g(qp),

where g (gp), as in preceding papers, denotes the expression -kr | 2π1.




It will be convenient to take the expression divided by a constant factor, defining the function F (qp) by the relation

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Dm-2 F (gp) = g(qp),

F itself being m-symphonic. In particular


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Green's Theorem for the operator D2.

§ 2. Take the ordinary form of Green's theorem

ди до


[za a dp = - fu do dt - fus'vdp,


(5) xx x

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in which is regular within the given region along with its first derivatives, and u, generally regular, may become infinite like 1/r; dp as in previous papers denotes an element of volume at the variable point p, and dt an element of the boundary at the point t. If in this equation we put v=g(qp), which is a function of r only, and surround the point q which is supposed within the region considered by a small sphere with centre 9 and radius p, we find in the limit when p decreases indefinitely

(6) 2u (q) = fu(t) h (tq) dt + for 3, 9 (pq) dp

ər ər


+ k2 fu (p) 9 (9P) dp.

If the pole q is outside the region, g (qp) is everywhere regular, and the first member of this equation must be replaced by zero. If, however, q is on the boundary, then in virtue of the property of a double stratum that the potential at a boundary point is the mean of the potentials immediately on either side, the first member of (6) becomes u (q).

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