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Taking real parts of (3) we have

logƒ' (a + iẞ)| = } { F (a + iß) +☞ (a − iß)}

log (A+B) = F (a + iß) + ℗ (a− iß). Differentiating with regard to a, B, and eliminating 'from the resulting equations, we obtain

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Now from the definition of A, B in terms of the modulus of a function they are necessarily real functions of a, B; we can therefore write down the values of P and Q at once. Further the expression on the left-hand side is an analytic function of a+iß.

The same is therefore true of the right-hand side and

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The former yields merely an identity; from the latter we have

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dA ď3B dB d'A

= 0................. (6). da d dB da3

Of the two functions da/da and dB/dB, it may happen that either or both are zero, or neither is zero. If both are zero A, B are mere constants and (5) is satisfied identically. From (4)

and from (3)

(a + iẞ) = constant,

x + iy=ƒ (a + iß) = C (a + iß) + D,

where C and D are constants. a, B are therefore merely Cartesian coordinates, and we obtain rectangular harmonics.

Let us next consider the case when da/da is zero, but dB/dB is not zero. From equation (5) we have

(A + B)



where A is now a constant.

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The constants c, d merely determine the origin and scale of measurement of a + is, and we may without loss of generality take

a + iß = log(x+iy),

a, B are therefore polar coordinates and we are led to circular cylinder functions or Bessel functions.

If neither da/da nor dB/dB is zero (6) may be written.

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One side of this equation being a function of a and the other a function of B, either side must be equal to a constant. Hence

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=-a ('+2c+e),


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'= a (A2+ 2bA+d), (d)'

where b, c, d, e are constants.

Substituting in (5) we have, provided that a is not zero, (b + c) (A − B) + (d− e) = 0.

Hence, since we have assumed that A, B are not both constants,

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c=-b and d=e,



= a (A2 + 2bA + d),

=— a (B' — 2bB+d).

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A + B = √(d − b2) {sinh√/a (a + h) + sinhi√a (ẞ + j)}. Substituting in (4) we have

F' (a + iß) = }√a


cosh va (a + h) + coshiva (3+j)
sinh Va (a+h)+sinh iva (3+)

= √a coth√a (a + iß + h + ij).

Đ (a + i3)=log sinh } Va (a+i3+h+j)+loge where c is a constant.


From (3)

f (a+i3) = c sinh ≥ Va (a + i +h+j)

ƒ(a+iß) = (2c| √a) cosh √a (a + iß + h + i).

Changing the scale of measurement and origin of a, ß we have, without loss of generality,

x + iy=f(a + iß) = c cosh (a + iß).

Thus a, are elliptic coordinates, and we obtain solutions of (1) in terms of elliptic cylinder functions.

If a is zero we have from (7)

A = b (a + c)2+d, B=b' (8+c')* + ď′

where neither b nor b' is zero, since we have already considered the case when either A or B is a constant.

Substituting in (5) we have

(b ́ − b) {b (a + c)' — b' (ß + c')'} + (b + b') (d + ď′ ) = 0.

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Again we may, without loss of generality, drop the constants, and we have

From (3)

and therefore


(a+iß) = log (a + iß).

f'(a + iẞ) = a + iß,

iy=ƒ (a + iß) = † (a + iß)',

a, B are therefore parabolic coordinates, and we are led to solutions of Laplace's equations in terms of parabolic cylinder functions.

Thus we see that cylindrical harmonics of the product type exist in Cartesian, polar, elliptic, and parabolic coordinates, but in no others.

University College, London.





1. THE algebraic surfaces which are multiply generated by conics were studied by Koenigs. He proved, subject to certain restrictions which I have elsewhere removed, † that such a surface is rational, and representable parametrically in the form

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The points of the section of the surface (1) by a plane Eux=0 are in (1, 1) correspondence with the points of the

*Annales de L'Ecole Normale Supérieure, ser. 3, vol. v., pp. 177–192 (1888). t." On sextic surfaces having a nodal curve of order nine," American Journal of Mathematics, vol. xxxvii., p. 447, cf. Art. 7.

binodal quartic curve Euf (, )=0. The genus of the section does not exceed unity. The order of the surface does not exceed eight, but this may be reduced by the appearance of fundamental points in the system of quartics Zu, f=0, additional to the fundamental double points at λ=∞ and μ=∞o. In this paper surfaces of order seven and eight, only, are considered. The surfaces of order six I studied in the paper cited above. The quintic surfaces of the given type have been the subject of several memoirs, the most extensive being by Caporali. The surfaces of the given type of order less than five all belong to well-known classes. We shall see that the surfaces of order seven and eight are generated precisely twice by conics. Those of lower order are all generated at least three times by conics.

I. Surfaces of order eight.

2. If the given surface is of order eight,† the curves Eu f=0, which correspond to the plane sections, have no fundamental points except the double points P, and P, at infinity on the X-axis and on the μ-axis respectively.

To the pencils of lines λ= const. and const. through P, and P, correspond the two systems of conics on the surface. In particular to P, corresponds a conic of the system λ = const. and to P, a conic const. Moreover, these two conics are in no way specialized on the surface, since a suitable birational transformation of the Au-plane will transform any pair of conics of the two systems into P, and P,.


The planes of each system of conics on the surface. envelope a curve of class six (Cf. however, Art. 18). Thus, the planes of the conics λ= const. constitute the system

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The sextic developables of the planes of the two systems of conies have twenty planes in common (Art. 8).

Annali di Matematica, ser. 2, vol. vii., pp. 149–188,

A special case of this surface was studied by Brambilla, Giornale di Mat, vol. iv. p. 1.

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