two legs of a tilting mirror, as in the other experiment. This was placed in a horizontal coil of elliptical cross-section. The demagnetizing force for a specimen of these dimensions is considerable, but the field and the magnetization near the horizontal scale. No. 1. Cyclic curve in low fields, well-annealed specimen. middle part are both sufficiently uniform. With this I obtained cyclic change of width curves of the reverse character to those for change of length, the changes being about half in amount; but the variation with different adjustments was so great that they are without value. I hope, however, to take up this subject again. These experiments were performed in the Cavendish Laboratory, and I am glad to be able to express here my most hearty thanks to Professor Thomson for the facilities and suggestions he has given me. XLVI. On Double-Refraction in Moving Viscous Liquids. By LADISLAS NATANSON, Professor of Natural Philosophy at the University of Cracow*. ACCIDENTAL double-refraction is produced in a tran sparent substance by distorting the latter, the doublerefraction disappearing as soon as the distorting cause ceases to act. Observed for well-nigh a century in the case of solids, it was discovered, in 1873, by E. Mach† in plastic bodies, and almost simultaneously-by Clerk-Maxwell in the case of certain highly viscous liquids. In 1881 A. Kundt § attacked the problem of double-refraction in liquids, and considerably advanced the subject. This physicist subjected the liquids under study to constantly re-applied and hence permanent distortions; in this way he was able to establish the phenomenon of double-refraction (also permanent) for a considerable number of liquids. In the same paper Kundt, starting from certain hydrodynamical theorems established by Sir G. G. Stokes ||, has attempted to develop the theory of the experimental method adopted by him. The merit of having carried out the first quantitative measurements in connexion with accidental double-refraction in liquids rests with G. de Metz T, who in his experiments made use of the principle employed by Kundt. The same method has been used by K. Umlauf**, J. E. Almy††, and Bruce V. Hill‡‡, and has enabled them to obtain relatively accurate results, in spite of the many difficulties inherent in this kind of research. Finally, in a recent paper by R. Reiger §§ will be found an account of some very interesting experiments on accidental double-refraction in certain plastic substances; the method adopted by Reiger approaches more nearly that which enabled Mach and Maxwell to demonstrate the reality of the phenomenon in question. In the present communication we propose to discuss the theory of accidental double-refraction in liquids; we shall, * Translated from the Bulletin de l'Académie des Sciences de Cracovie, March 1901. Communicated by the Author. + Optisch-akustische Versuche, Prag 1873. Proceedings of the Royal Society, No. 148 (1873); Scientific Papers, vol. ii. p. 379 (1890). § Wiedemann's Annalen, Bd. xiii. p. 110 (1881). Transactions of the Cambridge Philosophical Society, vol. viii. 1845; Mathematical and Physical Papers, vol. i. p. 102 (1880). ¶ Wiedemann's Annalen, Bd. XXXV. p. 497 (1888). ** Ibid. Bd. xlv. p. 304 (1892). + Philosophical Magazine (5) vol. xliv. p. 499 (1897). It Ibid. vol. xlviii. p. 485 (1899). $$ Physikalische Zeitschrift, Jahrg. ii. no. 14, p. 213 (1901). Phil. Mag. S. 6. Vol. 2. No. 11. Nov. 1901. 2 I however, confine ourselves to an examination of that part of the problem which one may hope to solve by purely hydrodynamical means; we shall not attempt to enter into considerations derived from optical theory, which would be necessary to render the theory of this phenomenon complete. § 1. Thanks to the investigations of which we have given a brief account, the particular case discussed by Stokes and realized in the experiments of Kundt has become of the greatest importance; it is therefore the case which will claim our attention. Imagine a cylinder kept in rotation. about its axis. At a sufficiently small distance from its surface imagine a fixed cylindrical wall having the same axis. The annular space bounded by the surface of the cylinder and the wall is filled with the liquid which it is desired to investigate. Let a internal, and b external radius of annular space, and r=distance from the axis of any arbitrarily chosen point M in the interior of the liquid. Let = be the velocity, and h the angular velocity of M. Let the axis of z be chosen along the axis of the cylinder, the axes of x and y being arbitrarily chosen in a plane normal to the axis. Let be the angle between the direction of r and the x-axis, and s=velocity of rotation of cylinder; this rotation will be assumed to be uniform. The motion communicated to the molecules of the liquid will evidently be along circles whose planes are normal to the axis of ≈. We shall assume that the velocity 9 of a molecule depends solely on its distance r from this axis: it is independent of the time t, so that the motion may be called steady. In practice, the actual motion of a liquid can only approximately fulfil the conditions of the ideal case which we shall be content to discuss here. From what has been said, it follows that the angle being reckoned from the instant t=0. The sign has been introduced in order to express the fact that the motion of the cylinder, and hence also that of the liquid molecules, may take place in two opposite directions relatively to the co-ordinate axes. In our calculations, it will sometimes be convenient to suppose that the liquid under consideration is incompressible; we shall further neglect the effect due to external forces, such as gravitation. After all, the part played by these auxiliary hypotheses in our reasoning will be a small one. § 2. The component of the velocity of M parallel to the axis of z is zero. The two other components are The double sign in these equations has the same meaning as in equations (2) and (3) in the preceding paragraph; it must, of course, be taken in the same consecutive order wherever it occurs. Denoting by F any function of the variables r,, we have The equations (2), along with (1), enable the components of the velocity of apparent deformation to be calculated. Among these components, those which we shall have to consider are the following: §3. In a previous paper (to which reference has been made) we gave the definition of what we term the true deformation. Let be the components, referred to the axes Or, Oy, Oz, of this deformation. We shall refer the same deformation to three new axes Ox1, Oy1, Oz, chosen as follows :-the direction of the axis O, is at every instant the same as that of the velocity g; the axis Oy, is coincident with r; and Oz, coincides with the axis of rotation, i. e. with the axis Oz. Referred to these axes, the actual deformation has for its components For the justification of this term the reader is referred to our preceding paper, “On the Laws of Viscosity," supra, p. 342. Among these components, the only one which we shall have to consider is yr. For calculating its value we use the well-known equation y* = 2€*cos (x1x) cos (y1x) +2p* cos (x1y) cos (y1y) + 24* cos (x15) cos (y +a*{cos (x,y) cos (yi) + cos (y1y) cos (x1z) +ß*{cos (x1≈) cos (y1x) + cos (y12) cos (x1x)} +y*{cos (x1x) cos (y1y) + cos (y1) cos (x1y)}. (3) If in this equation we put, in accordance with what has been said above, cos (x1x) = ± sin ℗ ; cos (y,x) = cos ✪; cos (≈1x)=0; (4a) cos (xy)=cos; cos (y1y)=sin ; cos (y)=0; (4b) cos (1)=1, (4) Y = ±2(e*—*) sin cos +y* (sin2 -cos). (5) § 4. The components of the actual deformation are capable of being expressed in terms of the components of the velocity of apparent deformation. We have, in fact, where krr, kyy, kry denote three constants, e the base of the naperian logarithms, T the time of relaxation, o the sum e+f+g, A* the sum e*+*+*. These equations may be established by the aid of considerations contained in our previous communication. Compared with the equations. which express the generalization of Hooke's law (as explained in the memoir referred to), they lead directly, in the case where it is permissible to put h=k, to equations (9) and (10), § 8 of the previous memoir. In equations (1), (2), (3), let us neglect the terms which contain kr, kyy, and key; then, using these equations along |