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are 3+,3+,3+, where the denominator we begin with is the first integer greater than the half of 7: similarly, that before we come to

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and so on.

3+

1 1

3+7+10'

When an even partial denominator occurs, we take as the partial denominator to begin with, either its half or the first integer greater than its half, according as the partial denominator following is greater or less than that preceding, or, these being equal, according as the next following is less or greater than the next preceding, and so on.

Another improvement, though verbal, is important, viz. in regard to the term convergent, the present definition of which seems arbitrary and unreasonable. With great convenience it may be defined as follows:-A convergent of a fractional number is a fraction which is a closer approximation to the given number than any other fraction with a smaller denominator; so that Lagrange's problem is simply to find all the convergents of any fraction.

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On the Relation between two continued Fraction Expansions for Series. By THOMAS MUIR, M.A., F.R.S.E.

On the Use of Legendre's Scale for Calculating the First Elliptic Integral. By Professor F. W. NEWMAN.

Denoting the first elliptic integral by F(c, w), and taking r such that

= F(c, w) : F(e, 1); then, in Lagrange's scale, from we deduce successively w1, W2, Wz.. by a given law, with the aid of c1, C2, previously determined from c. Then is the limit to which w, 2-1, 2-2w2) -3 @3 converge. If c is moderately small, the convergence is rapid. But if c is very near to 1, it may be expedient to reverse the direction of the new amplitudes and moduli, viz. to calculate e backwards c', c", c"", so as to make c"", c", c', c, C1, C2, a series continued by a single law; and similarly from o calculate backwards w', w", w''' Then w',

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....

w", w"" are proved to converge to a fixed limit o and F(c, ∞) : F(b, }π)=Nap log tan (+): . The function Nap log tan (+) involves but a single element, and was calculated by Legendre. Gudermann has since published a far ampler table. In practice the limit w' is quickly reached: often it suffices to make

and

w', at worst w=w". Thus for very large values of c2 Lagrange's scale practically suffices, presuming that we have at hand tables of F(c, d) and F(b, π). But Legendre, who discovered a new scale after completing his principal calculations, regarded his new scale as having much advantage in finding F(c, w) at once rapidly and accurately. In it is the limit of w, 3-1, 3-2, 3-33 the convergence, generally excellent in Lagrange's scale, is far more rapid in Legendre's. In Lagrange's scale the relation of o, to o is tan (w,-)=b'tan &. The relation in Legendre's scale is to the eye as simple, viz. tan (w,—w)=A tan w; but in the constant A, = (1-c2 sin 23), the value of 3 is determined by the equation F(c, ß)=3F(c, 3). A practical difficulty arose in the very considerable trouble needed to obtain A (or its logarithm) numerically when c was given. Legendre showed how ẞ was obtainable from c: the cubic equation arising can be solved by a mere extraction of the cube-root; but there are also two quadratics involving two extractions of the square-root. Then from ẞ we have to câlculate √(1—c2 sin 2ß)

from w.

and find its logarithm before we can proceed to deduce w All these operations have to be repeated to find w, from w; nay, we must first find c, from c, and that is still more arduous.

=

F(b, π) "F(c, π)'

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But when we assume p, π as argument, all is greatly simplified. The relation of c, C1, C2, C3 .... in Lagrange's scale corresponds with p, 2p, 22p, 23p, ...., and in Legendre's scale with p, 3p, 3'p, 33p...., which involve no trouble in calculating. No doubt we need tables (of single entry and easily compiled) to yield c, b when p is given, and p when c is given. Presuming these, we may treat a and F(c, w) as functions of p and w; after which the difficulties of the constant multiplier A vanish, and Legendre's scale becomes practical to us.

Denote log A, i. e. - log √(1-c2 sin 28), for the moment, by (p) (here the common log is intended); then, among the numerous series which express functions of the amplitude o in terms of ≈ and p, the author selects (with λ for Napier's log) 1. cos 2x 11- cos 6x 11- cos 10x + sin 2p sin 6p

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3

+5sin 10p

+ &c.,

); hence (p)

where sin p is written for (e-e). By hypothesis, F(c, B)=F(e, when wẞ, x=1, and we get, writing cosec p for the reciprocal of sin p, =M{cosec 2p+ cosec 10p+ cosec 14p+ cosec 22p+&c.}, M being the modulus of the common logarithms.

Assuming that we have a table of (p), then given p and w we have the equation log tan (ww)= log tan w-(p) to find w,; log tan (ww1)=log tan W1 -(3p) to find ; log tan (w-w)=log tan -(p) to find w, and so on. The approximation is sufficient when (3"p) is negligible; and this result is obtained so rapidly, that in the extreme case of p, x=3-2, is correct to ten decimals.

To bring the method to a practical trial, the author has calculated to twelve decimals a skeleton table of (p) for p=0·5, 0·6, 0·7, 0·8, 0·9, and from p=1 to p=143 at intervals of 0.1. The table is given in the paper, and also examples of the method. The process also by which the table was constructed, with the aid of tables of cosec p and e-P, previously calculated by the author, is explained.

General Theorems relating to Closed Curves. By Professor P. G. TAIT.

The closed curves contemplated are supposed to have nothing higher than double points. By infinitesimal changes of position of the branches intersecting in it, a triple point is decomposed into three double points, a quadruple point into six, and generally an æple point into (1) double points. (1) A closed curve cuts any

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infinite unknotted line in an even number of points [infinite here implies merely that both ends are outside the closed curve]. (2) The same is true if the line be knotted. (3) If any two closed curves cut one another, there is an even number of points of intersection. (4) In going continuously along a closed curve from a point of intersection to the same point again an even number of intersections is passed. (5) Hence in going round such a closed curve we may go alternately above and below the branches as we meet them. (6) By (3) the same proposition is true of a complex arrangement of any number of separate closed curves superposed in any manner. (7) In passing from the interior of any one cell to that of any other-in any system of superposed closed curves-the number of crossings is always even or always odd, whatever path we take. (8) Hence the cells may be coloured black and white in such a way that from white to white there is always an even number of crossings, and from white to black an odd number. Such closed curves therefore divide the plane as nodal lines do a vibrating plate.

The above are the enunciations of the propositions proved in the paper, which, with the necessary figures &c., will be found printed in extenso in the Messenger of Mathematics,' vol. vi., January 1877.

On a Theorem in the Mensuration of certain Solids.
By Professor JAMES THOMSON.

On Division-remainders in Arithmetic. By W. H. WALENN.

The author referred to a series of papers of his on unitation recently published in the Philosophical Magazine,' and to some remarks published in the Brit. Assoc. volume for 1870. If a divided by 8 leave remainder y, then the author calls y the unitate of a to the base 8, and writes Usay. The results of " unitation" may be conveniently applied to the verification of many numerical operations. The method of unitation is practically equivalent to the theory of congruencies, viz. the equation Uy would be written a=y (mod 8); and many of the results are identical with those given by Gauss,

On Many-valued Functions. By M. M. U. WILKINSON, M.A.

GENERAL PHYSICS, &C.

On the Transformation of Gravity. By JAMES CROLL, F.R.S.*

On the Influence of the Residual Gas on the Movement of the Radiometer. By WILLIAM CROOKES, F.R.S.

The author's recent experiments show that the movement of this instrument is not due to a direct repulsion exerted by light on the vanes, but to a mutual action called out between these vanes and the very attenuated gas remaining in the instrument. It is well known that, with a moderately good vacuum, the motion becomes more rapid as the exhaustion proceeds; but he has recently succeeded in producing such a complete exhaustion that he not only reaches the point of maximum effect, but goes so far beyond it that the effect nearly ceases. The vacuum is measured by means of a special apparatus, in which a moving plate, instead of continuously rotating in one direction, as in the ordinary radiometer, is suspended by a glass fibre, which it twists in opposite directions alternately. The movement is started by rotating the whole apparatus through a small angle, and the observation consists in noting the successive amplitudes of vibration when the instrument is left to itself, a mirror and spot of light being employed for this purpose. The amplitudes form a decreasing series, with a regular logarithmic decrement. The logarithmic decrement is nearly constant up to the point at which the vacuum is apparently equal to a Torricellian vacuum, the mercury in the gauge standing at the same height as a barometric column beside it; but as the exhaustion proceeds beyond this point, the logarithmic decrement becomes smaller-in other words, the amplitude diminishes less rapidly. By plotting the observations and supposing the curve continued, it is indicated that, if a perfect vacuum were attained, and the glass fibre had no viscosity, the logarithmic decrement would be zero, we should have perpetual motion with constant amplitude, whilst, at the same time, the radiometer would cease to act. Other gases as well as air have been tried. Aqueous vapour is very unfavourable to the action of the radiometer; hydrogen, on the contrary, gives the best result of all. Several experiments have been already described, which seem to point to the true explanation of the action of the radiometer; but the author thinks Mr. Stoney's explanation the clearest. According to this, the repulsion is due to the internal movements of the molecules of the residual gas. When the mean length of path between successive collisions of the molecules is small compared with the dimensions of the vessel, the molecules, rebounding from the heated surface, and therefore moving with an extra velocity, help to keep back the more * Printed in extenso in the Phit. Mag. 1876, ii. p. 241.

slowly moving molecules which are advancing towards the heated surface; it thus happens that though the individual kicks against the heated surface are increased in strength in consequence of the heating, yet the number of molecules struck is diminished in the same proportion, so that there is equilibrium on the two sides of the disk, even though the temperatures of the faces are unequal. But when the exhaustion is carried to so high a point that the molecules are sufficiently few, and the mean length of path between their successive collisions is comparable with the dimensions of the vessel, the swiftly moving, rebounding molecules spend their force, in part or in whole, on the sides of the vessel, and the onward crowding, more slowly moving molecules are not kept back as before, so that the number which strike the warmer face approaches to, and in the limit equals, the number which strike the back, cooler face, and as the individual impacts are stronger on the warmer than on the cooler face, pressure is produced, causing the warmer face to retreat*.

Mechanical Theory of the Soaring of Birds. By W. FROUDE, F.R.S.

On the Passage of Fluids through Capillary and other Tubes.
By Professor F. GUTHRIE and Dr. F. GUTHRIE.

On the Modification of the Motion of Waves produced by Fluid Friction. By Prof. J. PURSER.

On the Forces experienced by a Lamina immersed obliquely in a Fluid Stream. By Lord RAYLEIGH, F.R.S.+

On the Resistance encountered by Vortex Rings, and the Relation between the Vortex Ring and Stream-lines of a Disk. By Prof. OSBORNE REYNOLDS.

Description of the Bathometer. By Dr. C. W. SIEMENS, F.R.S.

On the Amplitude of Waves of Light and Heat,
By G. JOHNSTONE STONEY, F.R.S.

On Acoustic Analogues to Motions in the Molecules of Gases.
By G. JOHNSTONE STONEY, F.R.S.

Experimental Illustration of the Origin of Windings of Rivers in Alluvial Plains. By Professor JAMES THOMSON, LL.D., D.Sc.

The author referred to a communication which he had made to the Royal Society in the month of May last ‡, in which he had given a new theory of the flow of water round bends in rivers and round bends in pipes, and had explained the reason why, in alluvial plains, the bends of rivers go on increasing by the wearing away of the outer bank, and the deposition of mud, sand, and gravel on the inner

*For further researches on this subject, see papers read before the Royal Society, November 16, 1876, and on April 26, 1877.

+ Printed in extenso in the Phil. Mag. 1876, ii. p. 430.

Proc. Roy. Soc. vol. xxv. p. 5.

bank. The theoretical view which he had then offered, he now, for the first time, had verified by practical experiment; and this experiment he showed in the meeting. The chief point of the new view now experimentally proved was that the water in turning the bend exerts centrifugal force, but that a thin lamina of the water at bottom, or in close proximity to the bed of the river, is retarded by friction with the river-bed, and so exerts less centrifugal force than do like portions of the great body of the water flowing over it in less close proximity to the river-bed. Consequently the bottom layer flows inward obliquely across the channel towards the inner bank, and rises up in its retarded condition between the inner bank and the rapidly flowing water, and protects the inner bauk from the scour, and brings with it sand and other detritus from the bottom, which it deposits along the inner bank. The apparatus showed a small river, about 8 inches wide and an inch or two deep, flowing round a bend, and exhibiting very completely the phenomena which had been anticipated.

On Metric Units of Force, Energy, and Power, larger than those on the Centimetre-Gram-Second System. By JAMES THOMSON, LL.D., D.Sc., F.R.S.E., Professor of Civil Engineering and Mechanics in the University of Glasgow. The author premises that under the excellent method of Gauss for establishing units of force, a unit of force is taken as being the force which, if applied to a unit of mass for a unit of time, will impart to it a unit of velocity. In the system already adopted by the British Association Committee on Dynamical and Electrical Units (Brit. Assoc. Report, part i. 1873, page 222), the Centimetre, the Gram *, and the Second were taken as the units of length, of mass, and of time; and the unit of force thence derived under the method of Gauss was called the Dyne.

That force is very small, quite too small for convenient use in all ordinary mechanical or engineering investigations. It is about equal to the gravity of a milligram mass, and that force is so small that it cannot be felt when applied to the hand. That system, designated as the Centimetre-Gram-Second System, recommended by the Committee of the British Association, and described fully, with many applications, in a book since published by Dr. Everett, who was Secretary to the Committee, is well suited for many dynamical and electrical purposes; and it ought certainly to be maintained for use in all cases in which it is convenient. But the object of the present paper is to recommend the employment also of two other systems which are in perfect harmony with it, and to propose names for the units of force under these two systems.

In one of these systems, the Decimetre, the Kilogram, and the Second are the units adopted for length, mass, and time; and thus the system comes to be called the Decinietre-Kilogram-Second System.

In the other, the Metre, the Tonne †, and the Second are adopted as the units of length, mass, and time; and thus the system comes to be called the Metre-TonneSecond System.

It is to be particularly observed that all the three systems here referred to are framed so as to attain the condition, very important for convenience, that the unit of mass adopted is the mass of a unit volume of water, and that, therefore, for every substance the specific gravity and the density, or mass per unit of volume, are made to be numerically the same.

In the Decimetre-Kilogram-Second System, the unit of force derived by the method of Gauss is 10,000 Dynes, or is about equal to the gravity of 10 Grams. It is impossible, or almost so, to work practically with any such system without having a name for the unit of force. The unit of force in this system is such that a human hair is well suited for bearing it as a pull, with ample allowance of extra

*The spelling Gram, instead of Gramme, for the English word is adopted in the present paper in accordance with the spelling put forward in the Metric Weights and Measures Act, 1864, which legalizes the use of the Metric System in Great Britain and Ireland.

+ The Tonne is the mass or quantity of matter contained in a cubic metre of water, and is very nearly the same as the British Ton.

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