d't

dr2

[It should be remarked that the values of K' and K'. 10 have been calculated in the above tables for smaller intervals. (taken from the beginning to the end of the curve), since only in this way can it be seen whether the curve follows the same law over all its length. Naturally the experimental error in the calculation of K is through this considerably greater.] Each of the curves changes its curvature, has a point of inflexion, i, e. where =0 the curve changes its sign from positive to negative; the curves approximate asymptotically to the Tor+K axis (i. e. not to the To, axis, but to one which is parallel to it and is removed by K) as well as to the T. axis. The concave and the convex parts of the curve seem, as far as one can judge, to be logarithmic; but the total curve evidently cannot be a simple logarithmic one. Tables I. and II. show that the following equation holds good for the above curves:

ου

ου

log (t2- too+K)—log (t,—to+K) + log (t。—t1)

-log (t-t2)=K'(72—71).

From this the differential equation (1) given above follows, which means: the velocity of the reaction at the time is directly proportional to the remoteness of the heterogeneous system from the point of equilibrium, T-T, and to the surface of the solid (ice or salt) in contact with the liquid T-Tov (or generally to the surface of contact of the reacting parts of the heterogeneous system) + the constant K, which we shall call the instability constant.

At the beginning of the reaction, when 7=0, T=Tcv, we

i. e. 7 is finite, the reaction can take place at any temperature, Should there be no instability constant K in the equation, we should have instead of (1)

Integrated, this equation gives

1

K"T=

t-too
log
to-too to-t

+ const.

At the beginning of the reaction, when 70, T=Toe, we

i. e. the temperature can at no value of T rise in a finite time, and this is an evident impossibility. It is true, we could avoid this difficulty by making the assumption that the equation

is to hold good only for heterogeneous systems; i. e. that T cannot be put = Toe, since in this case the solid part of the system =0, and the system becomes homogeneous. This, however, would scarcely be quite satisfactory and justifiable in view of the fact that overcooled or supersaturated solutions. always crystallize out by themselves after a certain limit of overcooling or of supersaturation has been reached, though no solid is present in the system. On the contrary, the instability constant K shows that there is always an internal force present in an overcooled or a supersaturated solution (generally in all heterogeneous systems which are removed from the point of equilibrium) which is striving to drive the system from a state of instability into a state of reaction, which is ultimately to bring it to a state of equilibrium. This force is the greater the further the system is removed from the point of equilibrium. In the above equation K is taken as directly proportional to the remoteness of the system from the point of equilibrium T.-T. Since, however, the value of K is small, it cannot be decided with certainty whether the real form of variation of K with the amount of overcooling is not. another one. It is the passive forces in the system which hinder the force represented by K from setting the same into a state of reaction; as, however, K increases with the amount of overcooling or supersaturation, &c., K ultimately arrives at a value greater than those counteracting passive forces, so that after a certain limit the system must crystallize out by itself.

If we put in the integral equation (2) T=T, we have

i. e. at To the temperature of the liquid cannot vary in any finite time, the system is in equilibrium. If we differentiate (1) a second time we get

the point of inflexion is in the middle between T, and Tor-K (not between T. and Tor; this enables us to determine K graphically). Before

It should be remarked that in the above equation (1) To means the real point of equilibrium. Since our apparent point of equilibrium T' differs from the real point of equilibrium T, only by 0°.00002 or 0° 00004, and the point T' can be further brought as near to T, as desired, it is clear that it is T, and not the variable T' which here comes into consideration. This follows also directly from the meaning of the equation (1), which requires that when T.-T'=0 no more reaction should be going on in the system. When T' becomes constant it means, not that no reaction is going on in the system any more, but that the variation of temperature of the system because the reaction becomes equal and of opposite sign to the variation of its temperature under the influence of the surrounding medium. Only when the effect of the surrounding medium is =0 we have the real T。, i. e. no more reaction is possible in the system.

As to the velocity constant K' in (1), it is a function of many factors of the nature of the reaction, of the nature of the solvent, of the nature of the dissolved substance and of its concentration, of the temperature, of the velocity of stirring, possibly also of some other factors, such as the velocity of supercooling of the liquid before the reaction is

started, &c. The constant usually decreases with the increase of concentration. The velocity constants of ice separation and of ice melting are very great also in the case of canesugar; and we can very well assume that the constant remains the same in dilute solutions.

Some very interesting experiments have been made by Stephan. Though he did not arrive at the law of velocity of evaporation itself, he made the very interesting observation that the velocity with which the same quantity of liquid evaporates at constant temperature and surface varies with the pressure of the air or gas resting upon the liquid :

here V is the velocity of evaporation, p1 is the partial pressure of the vapour, P-p1 the partial pressure of the air or gas. The equation must evidently hold good for the velocity of condensation of the saturated vapour to a liquid when the pressure of the air or gas resting upon the liquid varies. Quite an analogous law is also to be expected for the velocity of separation of salts from supersaturated solutions, or for the solution of a solid in a liquid if an indifferent substance be dissolved in the solvent. In the latter case P-p1 will be the concentration or osmotic pressure of the indifferent substance and p, the solution pressure of the solid at the given temperature. The constant C, as is evident, must vary with the solvent. Moreover, we can say that Stephan's law and others analogous exist only because the velocity constant varies with the solvent of the gas or of the dissolved substance, and that pure water, e. g., and water containing an indifferent substance in solution are no longer the same solvent. (This also explains the deviations from Henry's law of absorption observed when indifferent substances are dissolved in the solvent.)

The analogy between the evaporation of a liquid or the sublimation of a solid, on the one hand, and the solution of a solid salt, on the other, allows us to establish the following relation, one not very well admitting experimental proof:The velocity of condensation of oversaturated vapour to a liquid or a solid is, at a constant surface of the liquid or of the solid, directly proportional to the degree of supersaturation. The velocity of evaporation of a liquid and the velocity of sublimation of a solid are, at a constant surface of the liquid or of the solid, directly proportional to the remoteness of the system from the point of equilibrium, or from the point of saturation. If the surface varies during the reaction, the velocity of reaction

T

at the time is further directly proportional to the surface of contact of the reacting parts at the time 7+ the instability constant K. At a constant surface of contact of the reacting parts of the heterogeneous system we have + K = constant, i. e. the instability constant does not become evident from the form of the equation.

Davy-Faraday Laboratory of the Royal Institution.

IV. Damping of the Oscillations in the Discharge of a Leyden-jar. By H. BROOKS, B.A., Tutor in Mathematics, Royal Victoria College for Women, Montreal *.

THE

HE method employed in the investigation of this subject depends on the partial demagnetization of a magnetized steel needle when placed inside a solenoid through which a leyden-jar discharge is passed.

The action of a rapidly alternating current on a magnetized steel needle has been investigated by Prof. Rutherford (Trans. Roy. Soc., June 1896), who has shown that the method can be employed as a simple means of comparing the intensities of high-frequency currents, and also as a means of determining the damping.

Erskine (Wied. Ann. vol. lxii. Oct. 1897) has employed such magnetized steel needles for measurements of the resistance of metals and electrolytes for rapidly alternating currents, and also for the determination of specific inductive capacities.

The object of this investigation was to examine in detail the damping of the electrical oscillations in leyden-jar circuits under varying conditions of spark-length, capacity, and

pressure.

The appearance of an air-break in the circuit connecting the outer and inner coatings of the jar when the discharge was passing, was examined by Feddersen by means of a rotating mirror in which the spark was reflected. He found that the image consisted of a series of bright and dark bands rapidly decreasing in breadth and intensity; and when a large resistance was placed in the circuit the image became a broad band of light gradually fading away in intensity.

This method was used by Trowbridge (Phil. Mag. vol. xxx. 1890, and vol. xliii. 1897), who photographed the image of the electric spark drawn out by the rotating mirror and measured the distances between the successive oscillations shown by the dark bands on the photograph. In this way he was enabled to make a comparison of the damping in a few * Communicated by Prof. E. Rutherford.