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NOTICES AND ABSTRACTS

OF

MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS.

MATHEMATICS AND PHYSICS.
Mathematics.

Address by G. G. Stoxbs, M.A., F.R.S. tfc, Lucasian Professor of
Mathematics in Hie University of Cambridge.

It has been customary for some years, in opening the business of tho Section, for the President to say a few words respecting the object of our meetings. In this Section, more perhaps than in any other, we have frequently to deal with subjects of a very abstract cnaracter, which in many cases can be mastered only by patient study, at leisure, of what has been written. The question may not unnaturally be naked—If investigations of this kind can best be followed by quiet study in one's own room, what is the use of bringing them forward in a sectional meeting at all? I believe that good may be done by public mention, in a meeting like the present, of even somewhat abstract investigations; but whether good is thus done, or the audience are merely wearied to no purpose, depends upon the judiciousness of the person by whom the investigation is brought forward. It must be remembered that minute details cannot be followed in an exposition vivd voce: they must be studied at leisure; and the aim of an author should be to present the broad leading ideas of his research, and the principal conclusions at which he has arrived, clearly and briefly before the Section. It is then possible to discuss the subject-matter; to offer suggestions of new lines of experiment, or new combinations of ideas; and such discussions and suggestions, it seems to me, are among the most important business of a meeting such as this. Any one who has worked in concert with another zealously engaged in the same research must have felt tho benefit arising from the mutual interchange of ideas between two different minds. Suggestions struck out by one call up new trains of thought and fructify in the mind of another; whereas they might nave remained barren and unfruitful in the mind of the original suggester. The benefit of cooperation is by no means confined to the carrying out, according to a preconcerted plan, of a research involving labour rather than invention; it is felt in a most delightful form in the prosecution of original investigations. In a meeting like the present, we have the oenefit of the mutual suggestions, not of two, but of many persons, whose minds are directed to the same object. The number of papers already in the hands of your Secretaries shows that there will be no lack of matter in this Section: the difficulty will rather, I apprehend, be to got through the business before us in the time prescribed. On this account the Section will, I hope, bear with me if I should sometimes feel myself compelled, in justice to the authors of papers which are placed later on our lists, to cut short discussions which otherwise might have been further prolonged with some interest.

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On Capillary Attraction. By the Rev. F. Basitforth-, B.D .

The theories of capillary action brought forward by Laplace, Young, and Poisson lead to the same form of differential equation to the free surface of a drop of fluid. During the last fifty years many attempts have been made to compare theory and experiment, but the results arrived at seem to be quite unsatisfactory. The experiments have generally been made by measuring the heights to which fluids rose in capillary tubes. The smaller the diameter of the tube, the greater is the elevation or depression of a fluid; but at tho same time it becomes more difficult to secure a bore of a perfectly circular section and a surface perfectly clean. Laplace attempted to test his theory bv comparing the measured thickness of large drops of mercury with their theoretical thickness obtained by an approximate solution of his differential equation.

After duly considering all the circumstances of the case, it appeared to the author that the forms assumed by drops of fluid, of small or moderate size, afforded the best means for testing the theory of capillary action. The drops of fluid may rest on horizontal planes which they do not wet, or they may hang below horizontal surfaces which they do wet. Extensive tables have been calculated, -which give the exact theoretical forms of all drops of fluid resting upon horizontal planes, as mercury on glass, within the limits of size to which it seems desirable to restrict experiments.

In order to determine the exact forms of drops of fluid, a microscope has been mounted so that it can be moved horizontally or vertically by micrometer screws provided with divided heads. In tin focus of the eyepiece are two parallel horizontal and two parallel vertical lines, .orming by their intersections a small square in the centre. The lines are purposely made rather thick in order that they may be seen without difficulty, and before reading off the screw-head divisions, care is taken to cause the image of the outline of the drop to pass through the middle point of the square caused by the intersection of the cross lines. Thus the coordinates are obtained of as many points as may be thought necessary, and afterwards the form of a section of the drop, passing through the axis of its figure, may be drawn by a scale of equal parts. By trial, a theoretical form must be fitted to this experimental form, using the tables. When this is satisfactorily accomplished, the value of Laplace's a is known, as well as the value of 6, the radius of curvature at the vertex: a determines the theoretical form of the drop, and 6 its size.

Only one or two satisfactory measurements have been made at present, but sufficient has been done to show that such values may be assigned to the constants aa to secure a most exact agreement of the theoretical with the experimental form of the free surface of a drop of fluid resting on a horizontal plane. It remains to bo seen whether a is constant for drops of all sizes of the same fluid at the same temperature. If experiment be found to agree with theory, then the effect of a variation of temperature upon a must be determined.

This method of proceeding affords the means of determining with great accuracy the angle of contact, because the tables calculated from theory give the coordinates for points, where the inclination of the tangent to the horizon is known, at intervals of one degree, and parts of a degree can De calculated for by proportioual parts.

If the experiments on mercury appear to confirm theory, it will be desirable to complete the tables for the forms of pendent drops of fluid, because it will be very difficult, if not impossible, to find supporting planes which such fluids as oils, water, spirit of wine, &c. do not wet or adhere to. In such case it appears to be possible to make use of pendent drops alone for the determination of n. When a has been determined for each of two fluids, as spirit of wine and oil, it will be desirable to examino the mutual action at their common surfaces, which may be done by measuring the forms of drops of ono fluid immersed in a bath of the other fluid contained in a cell having parallel and transparent vertical sides and horizontal planes at the top and bottom.

Since the differential equations of Laplace and Poisson are the same in form, it is evident that the above measurements for a single fluid cannot decide the difference between them. It seems, however, manifest that tho constitution of the surface is very different from the interior of a fluid, But the thickness of this surface of supposed variable density is so small as to bo insensible. Since there is a certain elastic force of vapour in contact with its fluid corresponding to every temperature, may we not assume that the density of this indefinitely thin envelope may vary from the density of the fluid inside to the density of the vapour outside?

On the Differential Equations of Dynamics. By Professor Boole, F.R.S.

Referring to the reduction^ by Hamilton and Jacobi, of the solution of the dynamical equations to that of a single non-linear partial differential equation of the first order, and to that, by Jacobi, of the latter to the solution of certain systems of linear partial differential equations of the first order,—the author showed, 1st, how, from an integral of one equation of any such system, a common integral of all the equations of the system could, when a certain condition dependent upon the properties of symmetrical gauche determinants is satisfied, be deduced by the solution of a single ordinary differential equation of the first order susceptible of being made integrable by means of a factor; 2ndly, how the common integral could be found when this condition was not satisfied.

On an Instrument for describing Geometrical Curves; invented by H. Johnston, described and exhibited by the Eev. Dr. Booth, F.R.S. This instrument supplies a want which has been felt by architects and sculptors. By its help, geometrical spirals of various orders may be described with as much manual facility as a circle may be drawn on paper by a common compass.

On a Certain Curve of the Fourth Order. By A. Catley, F.B.S. The curve in question is the locus of the centres of the conies which pass throuarh three given points and touch a given line; if the equations of the sides of the triangle formed by the three points are x=0,y=0, z=0, these coordinates being such that x+y+z=0 is the equation of the line infinity, and if ax+Py+yz=Qbe the equation of the given line, then (as is known) the equation of the curve is

»/az(y+z-x)+ */Py(z+x-y) + >/yz(x+y-z)=Q. The special object of the communication was to exhibit the form of the curve in the case where the line cuts the triangle, and to point out the correspondence of the positions of the centre upon the curve, and the point of contact on the given line.

On the Representation of a Curve in Space by means of a Cone and Monoid Surface. By A. Cayley, F.R.S.

The author gave a short account of his researches recently published in the 'Comptes Rendus.' The difficulty as to the representation of a curve in space is, that such a curve is not in general the complete intersection of two surfaces; any two surfaces passing through the curve intersect not onlv in the curve itself, but in a certain companion curve, which cannot be got rid of; this companion curve is in the proposed mode of representation reduced to the simplest form, viz. that of a system of lines passing through one and the same point. The two surfaces employed for the representation of a curve of the nth order are, a cone of the nth order having for its vertex an arbitrary point (say the point x=0,y=0, :=0), and a monoid surface with the same vertex, viz. a surface the equation whereof is of the form Qto —P=0, P and Q being homogeneous functions of (x,y,z) of the degrees p and p—1 respectively (where p is at most=n—1). The monoid surface contains upon it p (p— 1) lines given by the_oquationa (P=0, Q=0); and the cone passing through n(p Y) of these lines (if, as above supposed,/; ^> n —1, this implies that some of these lines are multiple lines of the cono), the monoid surface will besides intersect the cone in a curve of the nth order.

On the Curvature of the Margins of Leaves with reference to their GrowtJi.

By W. Esson, M.A. Leaves hare a right and left margin on each side of their axis. These margins

are of different lengths, but of tho same shape. The length differs owing to circumstances of growth, such as the left margin being next the stem or next a leaflet, forming with it a composite leaf. The curvature of the margin has been ascertained

in many instances to be that of the reciprocal spiral ^r=^. In some leaves

. the pole of curvature lies on the axis, in others in the body of the leaf, and in others entirely outside the leaf. If the leaflets of a composite "leaf have this curvature, their extreme points lie on a reciprocal spiral (e. g. the horse-chestnut leaf). It is probable that more irregular leaves have margins which are merely modifications

of the reciprocal spiral or other spirals, such as the Lituus ^r=-^=

The growth of a margin may be represented by increments of an arc of the spiral cut off by an increasing chord or radius vector. By this means may be accurately determined tho growth of a leaf under given circumstances of soil, temperature, and moisture. It is only necessary to register the amount of angular rotation of the radius vector of the spiral.

Quaternion Proof of a Theorem of Reciprocity of Curves in Space. By Sir William Rowan Hamilton, LL.D. 4'c Let <f> and ^ be any two vector functions of a scalar variable, and <f>, ^r, <f>", yfr" their derived functions, of the first and second orders. Then each of the two systems of equations, in which c is a scalar constant,

(1) . • . . S^-e, S^'f=0, S</>"^=0,

(2) . . . . S^=c, S^'<f>=0, S^"<j>=0, or each of the two vector expressions,

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includes the other.

If then, from anv assumed origin, there be drawn lines to represent the reciprocals of the perpendiculars from that point on the osculating planes to a first curve of double curvature, those lines will terminate on a second curve, from which we can return to the first by a precisely similar process of construction.

And instead of thus taking the reciprocal of a curve with respect to a sphere, we may take it with respect to any surface of the second order, as is probably well known to geometers, although the author was lately led to perceive it for himself by tho very simplo analysis given above.

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On a certain Class of IAnearYDifferential Equations. By the Rev. Robert Haeley, F.R.A.S. Theoijem.From any algebraic equation of the degree n, whereof the coefficients are functions of a variable, tliere may be derived a linear differential equation of the order n — 1, which will be satisfied by any one of the roots of the given algebraic equation. The differential equation so satisfied is called, with respect to tho algebraic equation, its " differential resolvent." The connexion of this theorem, which is due to Mr. Cockle, with a certain general process for the solution of algebraic equations, led the author to consider its application to the two following trinomial forms, viz.

y"-„y+(»-l)z=0, (I.)

y«_My»-i+(„_l)ar=0, (II.)

to either of which any equation of the nth degree, when n is not greater than 6, can, by the aid of equations of inferior degrees, be reduced. The several differential resolvents for the successive cases n=2, 3, 4, 5 were calculated; and by induction tho general differential resolvents were formed. Following Professor Boole's symbolical process and using the ordinary factorial notation, that is to say, representing

(»)(»-!) (n-2) . . . (n-r+1)

by [»]', the differential resolvent of (I.) was found to take the form

"IM^-(--i)^L^.i-^'-1^-wrWi]*-l«. •

In like manner, the differential resolvent of (II.) was found to be -1[(n-l)x|.]n-V-C»-l)(»x^-n-l)[»^-2]n-1^=[«-ir1x (B)

Every differential resolvent may bo regarded under two distinct aspects. It may be considered either, first, as giving in its complete integration the solution of the algebraic equation from which it has been derived; or, secondly, as itself solvable by means of that equation. In the first aspect the author has considered the differential equation (A) in a paper entitled " On the Theory of the Transcendental Solution of Algebraic Equations," just published in the ' Quarterly Journal of Pure and Applied Mathematics,' No. 20. In the second aspect every differential resolvent of an order higher than the second gives us, at least when the dexter of its defining equation vanishes, a new primary form, that is to say, a form not recognized as primary in Professor Boole's theory. And in certain cases in which the dexter does not vanish, a comparatively easy transformation will rid the equation of the dexter term, and the resulting differential equation will be of a new primary form.

On the Volumes of Pedal Surfaces. By T. A. Hiest, F.B.S.

The pedal surface being the locus of the feet of perpendiculars let fall from any point in space, the pedal origin, upon all the tangent planes of a given fixed primitive surface, will, of course, vary in form as well as in magnitude with the position of its origin. If, however, the volume of the pedal be considered as identical with that of the space swept by the perpendicular, as the tangent plane assumes all possible positions,—a definition which will apply to unclosed as well as to closed pedals,—the following two general theorems may be enunciated:—1. Whatever may be the nature of the primitive surface, the origins of pedals of the same volume are, in general, situated on a surfoco of the third order. 2. The primitive surface being closed, but in other respects perfectly arbitrary, the origins of pedals of constant volume lie on a surface of the second order; and the entire series of Buch surfaces constitutes a system of concentric, similar, and similarly-placed quadrics, the common centre of all being the origin of the pedal of least volume.

On the Exact Form and Motion of Waves at and near the Surface of Deep Water. By WrLLiAM Johw Macquobs BAincnrE, C.E., LL.D., F.R.SS. L. df E. 6fc.

The following is a summary of the nature and results of a mathematical investigation, the details of which have been communicated to the Royal Society.

The investigations of the Astronomer Royal and of Mr. Stokes on the question of straight-crested parallel waves in a liquid proceed by approximation, and are based on the supposition that the displacements of the particles ore small compared with the length of a wave. Hence it has been legitimately inferred that the results of those investigations, when applied to waves in which the displacements are considerable as compared with the length of wave, are only approximate.

In the present paper the author proves that one of those results—viz. that in very deep water the particles move with a uniform angular velocity in vertical circles whose radii dimmish in geometrical progression with increased depth, and consequently that surfaces of equal pressure, including the upper surface, are trochoidal— is an exact solution for all possible displacements, how great soever.

The trochoidal form of waves was first explicitly described by Mr. Scott Russell; but no demonstration of its exactly fulfilling the cinematical and dynamical conditions of the question has yet been published, so far as the author knows.

In 'A Manual of Applied Mechanics' (first published in 1858), the author stated that the theory of rolling waves might be deduced from that of the positions assumed by the surface of a mass of water revolving in a vertical plane about a

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