of the right-hand side of (16) equal to —A ̧+k'K sin 0; so that if N is the conical angle subtended at P by the circle, or the magnetic potential per unit current in the circle, we have the very simple expression Q=2A,—2k ́K sin 0. (18) Again, supposing that the depth, 00′ (fig. 1), of a coil consisting of a single series of circular currents is small compared with the distance of the point P from any part of it, the two terminal plates, ACB, A'C'B' may be considered as close together, and the potential of the coil at P is the value of V in (16) minus the value obtained by putting z+h for 2, where h=00'. Hence the potential in such a case is -mh, i. e., at any point in space whose distance from every part of a coil is great compared with the depth of the coil, the potential is 2mh (A.-KK sin 0). (19) The modulus k which appears in these equations, being is constant at all points for which p is constant; p2 ρ i. e., at all points on any circle which cuts that described on AB as diameter orthogonally. The circles which cut this latter orthogonally, having their centres on AB, are most readily drawn by joining A to points, m, n, p, (fig. 3) on the perpendicular at B to AB, and drawing perpendiculars at m, n, p, ... to Am, An, Ap, ...; the points of intersection of AB produced with these perpendiculars are the centres of the circles. If C is the centre of the orthogonal circle through this circle is p/2 p2, m, we know that the constant, 1- on AB AC' The field due to the plate AB is most readily mapped out by describing a large number of very close circles of the orthogonal system for a regular gradation of the values mAB, nAB, pAB, ... of ß, drawing a line BP in the assigned direction 0, and from Legendre's tables of Elliptic Integrals taking out the values of K, E, K, E'。. The properties of the orthogonal circles lead to some simple results with regard to potentials. Thus, if any line, AP, is drawn from A cutting any circle of the series in P and P', the lines joining P and P' to B are equally inclined to AB, i. e., LABP'=π-0. Now if in A, we put π-0 for e, we have, in virtue of Legendre's relation between complete complementary integrals, Aπ-▲..., i. e., Hence, from (18), if , ' are the conical angles subtended at P, P' respectively by the circle (or plate) AB, we have the remarkable relation Again, if V, V' are the potentials at P, P' due to the plate, we have from (15) V VI (23) a result which enables us to lay down the field at all points to the right of the perpendicular Bp when the field t left of Bp is known. Supposing now that instead of a single wire of diar AB, we have a series of wires forming a coil containe tween the diameter AB and the diameter ST, i. e., the breadth of the coil is BT or AS; then in calculating the potential at P we shall have to find the potentials due to a series of circular plates, each of surface-density m, and to add these potentials together. But observe that the potential at P due to any plate, AB, of radius a is of the form where (k, 0) is the coefficient of a in (15), and ø(k, 0) is a function of and e', the angles PAB and PBA. Hence if we take a plate of radius OQ, and from B draw Bq parallel to QP and meeting OP in 9, the potential of this plate at P is to the potential of the plate AB at q as OQ is to OA; for, if AR=BQ, the angles qBA and 9AB are equal, respectively, to PQR and PRQ. Hence, if r=OQ and V, is the potential at 9 due to the plate AB (of radius a), the resultant potential at P due to the series of plates of radii extending from OB to OT is the points q on OP ranging from t to P, where Bt is parallel to PT. Of course any plate of the series may be taken instead of AB as the reference plate. Thus, the resultant potential, due to all the plates, is calculated from values of the potential of any one plate at a series of points ranged along the radius vector OP. Pass now to the consideration of the practical problem in hand, viz., the potential at P due to a coil of depth OO', i. e., we have to consider the whole of the spaces BTT'B' and ASS'A' filled with wire traversed by a current of strength i. We have already seen that we have to subtract from the potential at P due to a series of uniform attracting plates, each of surface-density m, ranging from the radius OB to the radius OT, the potential at P due to the lower series, each of surface-density m, and ranging from radius OB' to radius OT'. It merely remains to express m in terms of current-density. If C is the total quantity of current traversing (at right angles to the plane of the paper) a unit area (square centimetre) of the space BTT'B', the quantity flowing in a filament of depth dy and breadth dr is Cdydr. Now this filament is replaced by the magnetic shell of radius r and thickness dy; and since we know that the strength of the shell is equal to the current in the filament, we have m. dy = Cdydr,..m=Cdr; hence (25) becomes, from (15), which is the potential due to the upper series of plates, OB,... OT. This quantity may be graphically represented and calculated as follows. Let a very close series of curves representing Phil. Mag. S. 5. Vol. 37. No. 225. Feb. 1894. Q a series of constant values of the function (k,) for the plate AB be drawn; draw OP, and at each point, T, Q, ... B, of the breadth BT of the coil draw an ordinate, Th, Qg,... Bf (fig. 4), equal to the product of r and the value of (k,) at the corresponding point, t, q,... P, of the line OP these ordinates will form by their extremities a curve, hg...f, the area of which multiplied by four times the current-density in the space occupied by the coil is the potential at P due to the upper series of plates, OB, OQ,... OT. 1 Now if we take the point P1 such that PP, is equal and parallel to OO', the depth of the coil, the potential at P due to the lower series of plates, OB',... OT', is equal to that at P1 due to the upper series. Hence, if A and A1 are the areas of the curve hg...f and the corresponding curve for the point P1, the total potential at P due to the complete coil is The curve hgf passes, of course, through the point O; and when the line OP coincides with the axis, 00', of the coil, the curve is an hyperbola; for, in this case We may, if we please, express r in terms of (k,0), and draw the curve hgf by a different rule. Thus, and we can make the ordinate of the curve equal to XVII. On the Thermal Behaviour of Liquids. To the Editors of the Philosophical Magazine. DUR URING recent years several papers on the thermal properties of gases and liquids, which contain what we believe to be incorrect conclusions, have been published; and in the interest of accurate knowledge we feel bound to make some comments on them, and to point out in what respects they are erroneous. We shall first consider certain observations on the critical point. Battelli (Atti del R. Istituto Veneto di Scienze, iv. serie vii. 1892-93; also a pamphlet published by Antonelli, Venice) comes to the following conclusions regarding the critical point: "1. The critical temperature is that at which the cohesion of the liquid particles is so diminished that they no longer remain together, but expand throughout the containing vessel. "2. Above the critical temperature the liquid particles continue to vaporize, i. e. to separate into molecules of saturated vapour, as the temperature rises. "3. Retaining for the term 'critical point' the signification which it has in the isothermal diagram, the determination of the critical point (temperature?) by means of the optical method is not generally exact; because the disappearance of the meniscus takes place at a temperature higher than the critical temperature, and striæ (intorbimento) appear at a temperature the lower, the greater the amount of liquid contained in the experimental tube." Again, in Wiedemann's Annalen, vol. 1. p. 531, Galitzine states that the critical temperature, determined by heating the liquid and allowing it to cool until a meniscus is just visible, is lower than the true critical temperature. He also makes the remarkable assertion that at temperatures considerably higher than the critical temperature the substance at constant pressure may have different densities, varying as much as 25 per cent., owing to the formation of liquid complexes! A short account of these conclusions is to be found in Nature,' November 23, 1893, and in this Journal, Dec. 1893, p. 552. Zambiasi and de Heen also state that the temperature at which mist and striæ appear when the substance is cooled is lower than the true critical temperature, but in other respects |