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where a1, a, a, are scalars, which separately vanish when the systems are isothermal.

Expanding the last equation we have

Vu1 gig1 + ▼q. q'. gig1 + gig ̄1▼q. q1 − 28 (gig ̄1) q. q' = a

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a1

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We obtain Dupin's Theorem in its most general form by operating by S., S.j", S.k' on this and the two similar equations respectively. It is thus expressed as three equations, of which one is S.ï' S (ï'▼) q. q ̄1=0.

Again, by multiplication by ", and by adding the other two equations multiplied by j' and ' respectively, we obtain also

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When the systems are isothermal as well as orthogonal, this equation may be put in the singular form

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The results given in this section were laid before the Society in May 1876, but were mislaid, with other papers then read.

(8.) The great desideratum in the application of quaternions to problems such as those just treated, seems to lie in the discovery of the general solution of the equation

where r is a quaternion.

vr = 0,

Unfortunately this seems to depend ultimately upon Laplace's equation, treat it how we may. It is

easily seen to be equivalent to the kinematical problem of finding a displacement which shall produce no compression, but shall produce a rotation whose vector axis itself corresponds to a displacement without compression.

The nature of the difficulty is also easily seen in another way; for, when we try to find the conditions of integrability of such an equation as

V. λάμπαν,

we may, of course, make the assumption

ἀμ = φαρ

where the coefficients of & are functions of p. This gives at once

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whatever constant vector be a.

Suppose this satisfied, we have the farther condition

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Taken in conjunction with the former condition, this shows that ▼ may here be considered as operating on A only.

In this very particular case, however, we find at once that A must be constant, and that

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OF THE

ROYAL SOCIETY OF EDINBURGH.

VOL. IX.

1877-78.

No. 101.

NINETY-FIFTH SESSION.

Monday, 7th January 1878.

The Right Rev. BISHOP COTTERILL, Vice-President, in the Chair.

The following Communications were read :—

1. On Gladstone's Theory of Colour-Sense in Homer. By Professor Blackie.

2. Note on a Geometrical Theorem. By Prof. Tait.

In "Trans. R.S.E." (1864-5) Fox Talbot proved very simply, by means of a species of co-ordinates depending on confocal conics, the following theorem, at the same time asking for a simple geometrical proof.

If two sets of three concentric circles, with the same common difference of radii, intersect one another-the chords of the arcs intercepted on the mean circle of each series by the extremes of the other are equal.

A properly geometrical proof may possibly be obtained by showing that the middle points of these arcs are equidistant from the line joining the centres. It is, of course, quite easy to build up a quasi-geometrical proof, but Talbot's would be much better.

The following investigation shows the nature of the theorem, and gives some elegant constructions.

Let d be the common difference, b and c the mean radii, and a the distance between the centres. Then the square of one of the chords is easily seen to be

VOL. IX.

p2 = 2c2 (1 −
- cos (0'+0)),

4 B

where and are given by

(b − d)2 = a2 + c2 - 2ac cos 0,

(b + d)2 = a2 + c2 - 2ac cos 0'.

The expressions

for the other chords differ only by the interElimination gives at once

change of b and c.

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=

1/2{4(a2 + d2)(b2 + c2) − 2(b2 — c2)2 – 2(a2 — d2)2

-

2(4a2c2 − (a2 + c2 − (b − d)2)2)1 (4a2c2 − (a2 + c2 − (b + d)2)2); }

4

=

- (A2 + A2 + 2AA')

F

where A and A' are the areas of the "inscribable" quadrilaterals, crossed and uncrossed, whose sides are a, b, c, d. This, of course, proves Talbot's theorem.

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Or,

a remarkably simple expression. The two values of p are given at once by Talbot's diagram, and the rectangles under their quarter sum, and difference, respectively, with the distance between the centres, give the areas of the quadrilaterals above mentioned. better, the triangles whose angular points are the middles of the arcs respectively, and the centres, have areas equal to half the sum and half the difference of the quadrilaterals.

The symmetry of these expressions shows that in Talbot's theorem any two of the four quantities employed may be interchanged-the lengths of the corresponding pairs of equal chords being always inversely as the quantity chosen for the distance between the centres of the two series of circles.

Again, it is easy to see that we have by the above equations

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so that, construct the figure how we will with four given lines, the ratio of the tangents of the halves of the pair of angles corresponding to 0, 0', is constant. This is the relation between True and Excentric Anomaly. And we have also the very simple expression

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so that the product of the areas of the crossed and uncrossed quadrilaterals is equal to the product of the areas of the (construction) triangles whose sides are

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respectively. Here again the letters may be interchanged at will; which, in itself, is a curious theorem.

While seeking a quaternion proof of the above theorem, I hit upon the following result. Given two opposite sides of a gauche quadrilateral in magnitude and direction. If one of these be fixed, and if the diagonals are to be of equal lengths, the locus of either end of the other is a plane.

Professor Tait, in consequence of the lateness of the hour, postponed his paper "On the Strength of the Currents required to work a Telephone." He said that the title given in the billet did not fully describe the contents. These referred not only to various measurements of the actual currents employed, whether produced from a cell or a Holtz machine, or by induction, but also to the mode in which the sounds are reproduced. He stated that he believed it would soon be possible to employ the instrument for the study of internal changes of form in all bodies, and also that in its construction magnets might be entirely dispensed with. He also stated that Mr James Blyth had with success substituted a copper plate for the

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