the conditions which must be satisfied in order that a binary quantic of this degree 2n may be a perfect square, and show that they may be all found from a matrix which I call the square matrix for the functions of the degree 2n. I have not entered on any discussion of these curious conditions and their intimate relationship, which are well worthy of examination, insomuch as their number is the number of ways in which 4n-3 quantities may be taken 2n-2 together, and are still equivalent to but n conditions. The Abelian system of differential equations may be written where there are m quantities Z1, Z2, Zm, and m−1 equations, as is clear from the above method of writing them if we suppose that i can have any integer value from i=0 to i=m−2; also ƒ (z) = z2m + P ̧z2m−1 + P ̧22m−2 + 2m-2 P2m I now form a function which I call F(2), and which is of the degree 2m-2 in the following manner. Now I say this function F (2) must be a perfect square. Forming, then, the vario conditions from the square matrix of F (≈), we have all the forms of the algebra integrals of the Abelian system which are rational and integral, involving m−1 arbitrary constants λ, Àg λ 2. On a Graphical Transformer. By A. P. TROTTER. This instrument is intended for the expeditious replotting of a curve wit transformed ordinates without calculation or scaling. It consists of a rectangula frame and a curved template or cam, and is used in conjunction with a straigl ruler. Let the scale of one system of ordinates be set off upwards along the edge‹ one of the perpendiculars, and the scale of the other along the edge of the othe perpendicular, but downwards. Join the corresponding points on the scale straight lines. The envelope of this system of lines may be thus drawn, and to th curve a cam is cut in thin wood or ebonite. To transform any ordinate, set the frame against a T square, adjusting the edg to the ordinate, and the zero to the zero of the scale. Set a needle at the extremit of the ordinate; bring a straight edge to touch the needle and the cam; prick o a point at the intersection of the straight edge with the other edge of the frame This point determines the length of the new ordinate. An instrument provided with a logarithmic cam was exhibited. With this instrument the product or quotient of two curves can be found by adding or sub tracting the logarithms of the ordinates; or the logarithms of a series of observations can be plotted. Cams for other functions can be easily made; but it must be 'Printed in extenso in the Electrician, August 17, 1894, vol. xxxiii. p. 465. remembered that the action of the instrument is, as it were, arithmetical rather than geometrical, for a cam is useful only with reference to its own scale. This instrument not only enables transformations of a definite and known character to be made, but is equally applicable for transforming in an empirical manner. The curve drawn by a recording voltmeter or ammeter may thus be replotted for estimation of area, or other graphical analysis, without any knowledge of the law of the particular instrument. In other words, a correction can be applied to a curve. The cams are easy to make, and even if carelessly cut cannot possibly give rise to cumulative errors. It is convenient to use the upper edge of the ruler instead of the edge which rolls on the cam. The curve must in this case be set out with the ruler, and used with the same ruler, or one of the same width. The rolling of a straight edge on a cam has been used in a photometer, invented by Mr. W. H. Preece and the author,' for the automatic calculation of the squares of the displacements of a lamp. 3. On a Linkage for the Automatic Description of Regular Polygons. By Professor J. D. EVERETT, F.R.S. Let any number of equal bars be jointed together in the manner of a lazytongs, so as to lie in two superposed planes, each bar in one plane (except the end bars) being jointed at both ends, B, D, and at one intermediate point, C, to the corresponding points of three bars in the other plane; but instead of the two distances BC, CD being equal, as in the ordinary lazy tongs, let them be unequal, CD being the greater. All the bars are to be precisely alike. They will form a frame with one degree of freedom, resembling in this respect an ordinary lazy tongs; but instead of the three series of points, B, B2. D1 D2. being ranged in three parallel straight lines, they will be ranged in three concentric circular arcs, two of which, namely, B1 B2 and D, D2 formed by the inner and outer ends respectively, will subtend the same angle at the common centre O. In place of the rhombuses of the ordinary lazy tongs we shall have kites, and the axes of all the kites will pass through O. When the frame, supposed to be at first pushed close in, is gradually opened out so as to increase the widths and diminish the lengths of the kites, the curvatures will increase in a double sense: the arcs formed by the inner and outer ends will increase in length, and at the same time their radii will diminish. The common angle which they subtend at the centre will accordingly increase very rapidly, and may easily amount to 360° or more. When it is exactly 360°, the first and last points B will coincide, as will also the first and last points D. In this position the points D will be the corners of one regular polygon, the points C of another, and the points B of a third. We have thus an automatic arrangement for constructing a regular polygon with any number of sides. Also, as the axes of successive kites are equally inclined to one another, we have the means of dividing an arbitrary angle into any number of equal angles-an end which can also be attained by employing the principle that equal arcs in a circle subtend equal angles at a point on the circumference, or, still more conveniently, by making use of the fact that those bars which correspond to parallel bars in an ordinary lazy tongs are equally inclined each to the next. Strictly speaking, the figures obtained are not polygons, but stars, which can be converted into regular polygons by joining the ends of their rays. It frequently happens that the curvature can be extended far beyond 360°, giving a succession of regular stars with a continually decreasing number of rays. Let each of the bars above described be lengthened at its inner end, B, till a point A is reached, such that the three distances, AB, AC, AD, are in geometrical progression. Then it can be shown that the radius OA of the circular arc formed by the ends A is constant, and equal to AC. Hence the common centre, O, can be found automatically by employing two additional bars of length AC, jointed 1 Proc. Inst. C.E., vol. cx. p. 81. together at one end, O, and jointed at their other ends to two of the points A. These bars OA may be called radius bars, the other bars, AD, being called long bars. The proof is easily gathered from an inspection of the accompanying figure, in which OA, OA,, OA,, OA are radius bars, A,D, A,D long bars, and A,C, A,C, portions of two other long bars whose remaining portions are indicated by dotted lines. The figure contains two equal and similar jointed rhombuses, OC,, OC,, and three similar kites, OB, DB, DO. Each of the bars A,C,, A,C, is cut in a fixed ratio at the point of crossing, B, the ratio of the smaller part to the whole being OA, : A,D; hence we can have joints both at B and D, as well as at O, and the other corners of the rhombuses, without hampering the motion. Two radius bars are in general sufficient to give the centre, but we may in theory attach a radius bar at each point A, and joint their other ends together at one point, O, which will be the common centre. There are difficulties in the way of realising this design in practice, except with a very limited range of movement; but by carefully selecting the best order of superposition of the bars, and by thinning off the radius bars towards the end where they are all superposed, it has been successfully carried out in two of the frames exhibited, each consisting of ten long bars and ten radius bars. The other frame exhibited illustrates the first paragraph of this abstract, and consists of ten bars BD. The number of bars in this frame might be increased indefinitely. If m denote the ratio of a long bar to a radius bar, or of the longer to the shorter sides of a kite, 2 a the angle between the two shorter sides, and 23 the angle between the two longer sides of a kite, then, by considering one of the two triangles into which a kite is divided by its axis, we have sin a/sin B = m, which is equivalent to tan (a-3)/tan (a + B) = (m-1)/(m + 1). a+ẞ is one of the angles of a rhombus, and a-ß is the angle between two consecutive rays of a star. In the case of the frame first described, consisting of 2n bars BD, when a regular star of n rays is formed, the central figure in the frame will be a polygon of 2 n sides, whose angles are alternately a + ß and 360° – 2 a. For the radii of the circular arcs we have (calling a radius bar unity) OD = √ { m2 + 1 + 2 m cos (a + B) }, OA is a side of a rhombus, and OC one of its diagonals. OD is the length of the largest kites, and OB the length of the inner kites. Auy two of these four radii may be equal, except that OD is always greater than either OC or OB. When OA=OB we have 2 cos (a+B) = -1/m: 2a+B=180°, cos a = sin † ß = 1/(2 m). OA = OC gives a + B = 120°. = - sin ß sin a; whence Of the two twenty-bar frames exhibited, one has joints at both B and D, with m = 2; the other has no joints at B, and its ends can be made to overlap so much as to give a three-rayed star. 4. On the Addition Theorem. By Professor MITTAG-LEFFLER. Professor Mittag-Leffler called attention to the intimate relation which exists between the modern theories of ordinary non-linear differential equations and the addition theorem. He explained how the theories created by Fuchs, Poincaré, and Picard may be generalised by making use of the considerations introduced by Weierstrass, and showed the direction which this generalisation must take. He also pointed out that the addition theorem itself may be generalised to a very considerable extent, and that the resulting theory has important applications to the theory of differential equations. 5. Note on a General Theorem in Dynamics. By Sir ROBERT BALL, F.R.S.. The following general theorem establishes a relation which characterises the particular type of screw-chain homography which is of importance in dynamics. Let a, B, y, &c., be a series of screw-chains about which a mechanical system of any kind with any degree of freedom can twist. Let η be the impulsive screw-chain which, if the system were at rest, would make the system commence to move by twisting about a. Let & be the corresponding screw-chain related to B, and to y, &c. Then the two systems of screw-chains, a, ẞ, y, &c., and ŋ, §, S, &c., are homographic. But this homography is not of the most general type. It was only lately that I succeeded in ascertaining the further general condition that the screw-chains must satisfy. Let denote the virtual coefficient of the screw-chains a and §; i.e., let this symbol denote the rate at which work is done by the unit of twist velocity about a against the unit wrench on έ. Then every three screws a, B, y in one group are connected with their three correspondents 7, έ, in the other group by the relation 6. The Asymmetric Probability Curve. By F. Y. EDGEWORTH, M.A. The asymmetric probability curve is the general form of the law of error. It may be obtained by solving a system of partial differential equations, which is the generalisation of the system given by Mr. Morgan Crofton for the symmetrical probability curve (Encyclopædia Britannica,' article on Probabilities, p. 781, equations 45, 46). The generalised system may be written: (3) = 1 day where y is the frequency with which any error a occurs; r is measured from the centre of gravity of errors; k is the sum of squares of errors measured from that point; the similarly measured sum of cubes. The solution of the system is a series of ascending powers of x, each term of which consists of a series of ascending powers of j÷k Ifj is put =0, the curve being treated as symmetrical, the series reduces, as it should, to the ordinary probability curve Ifj÷k is small, the curve being only slightly asymmetrical, the series reduces to a curve which is indicated by Todhunter as being related to the ordinary probability curve as a second is to a first approximation (Todhunter, 'History of Probabilities; Laplace, art. 1002, p. 568). This slightly asymmetrical probability curve may be used to correct the theory of correlation investigated by Messrs. F. Galton and H. Dickson ('Proc. Roy. Soc.,' 1886), and by the present writer ('Phil. Mag.,' November and December 1892). Whereas, according to the first approximation, the most probable y corresponding (or relative') to any assigned (or subject ') value of a lies on a right line passing through the origin, according to the second approximation the locus of correlates is a parabola. 7. On the Order of the Groups related to the Anallagmatic Displacements of the Regular Bodies in n-Dimensional Space. By Prof. P. H. SCHOUTE. 1. The groups related to the regular bodies in ordinary space have been amply studied by F. Klein in his Vorlesungen über das Ikosaeder' (Leipzig, Teubner, 1884). There in every case the order of the group has been found by enumeration of the possible positions; so the remarkable fact that this order is always twice the number of edges is not observed. I now wish to publish a simple general principle by means of which the order of the group may be easily determined. This principle will prove to be capable of immediate extension to n-dimensional space. 2. General Principle.-The manner of coincidence of the regular body ABCD with the given position PQRS . . . (say the orientation of ABCD with respect to PQRS) is determined if the position of the vertex A and that of the edge AB have been assigned. Therefore the order of the group is the product of the number of vertices by the number of edges passing through a given vertex. This product is evidently the twofold of the number of edges. Four-dimensional Space (S1). 3. General Principle extended.--The orientation of the cell ABCD . . . with reference to the given position PQRS . is determined, if the position of the vertex A, that of the edge AB through A, and that of the face ABC through AB |