rapidly with the distance*. The law indeed forms a limiting case for which the field of force surrounding each ion is infinitely strong but confined to an infinitely thin shell. This means that the mass law (4) applies solely to association and dissociation which is the result of what we may call "chemical" forces, using this name to distinguish forces of this type from the electrical forces of the free ionic charges which fall off very slowly with the distance. If association is produced by these (4) will not apply to it. (2) The preceding conclusion suggests that the failure of the mass law may be due to a mistaken view of the nature of the interionic forces which cause association. The apparent association may be partly-or wholly-due to the electric forces. In attempting to investigate this view the first difficulty is to know exactly what is the law of force between the ions. Are we to assume that every + ion attracts every ion and repels every + ion in the liquid according to the inverse square law? If the mixture of ions were a gaseous one this assumption would presumably be a sound one, but its validity is more doubtful in the case of an electrolyte, where the forces between the ions are affected by the intervening water molecules. As the first step, however, it seems the most straightforward assumption to make, and in previous papers † I have worked out by what is, I think, a strict method, the approximate effect on the osmotic pressure of a mixture of ions in which interionic force of this character is assumed to exist. The calculation, as might be expected from the complexity of the forces, is lengthy and need not be further referred to. The net effect of the interionic forces was found to give a reduction of the osmotic pressure which appears to be in accurate agreement with the experimentally found results for dilute aqueous solutions of strong binary electrolytes. In these cases therefore there is ground for believing that "chemical association," if existent, is extremely small, and that the effects observed are due entirely to the electrical interionie forces. Before we can apply to this view the proposition of the present paper, it is necessary to settle the relation in which the free ionic pressure p stands to the measured osmotic pressure P. Unless they are different from each other, an inequality * This is the kinetic aspect of the thermodynamical stipulation that the osmotic pressures of the ions must obey the perfect gas law. The assumption is formally made in Boltzmann's original deduction, and it can easily be shown that any other will result in a law different from (4). + Phil. Mag. xxiii. p. 551 (1912); xxv. p. 742 (1913). between the freezing-point and conductivity variations will result from the proposition. For if is the fractional alteration of pV, and therefore of u, and if further p=P, we shall get which is in conflict with the experimental result (3) (Part I.). p and P, however, cannot be the same, as is evident from the following consideration :-Consider a pair of ions which happen to be fairly close together and under the influence of each other's attraction, and let a number of representative views be taken. In a certain fraction-e1/T_of the cases the ions contribute to the free ionic pV, in the remainder of the cases the ions act as though bound together and contribute nothing to pV, but in these cases the pair will make to the measured PV exactly the same contribution as if the ions formed an actual molecule. The effect of electrical forces is in this respect exactly similar to that produced by chemical. association. The electrical bond it is true persists when the ions are well separated from each other, while the chemical bond acts only at very small distances, but this difference is immaterial in considering the molecular pressure which a pair of bound ions will exert. For the type of force considered, however, the bonds are not confined to single pairs of ions, but each pair must be considered as forming part of a large group, with the rest of the ions in which it possesses a certain mutual energy, and so is not free to exercise its full molecular pressure. While a difference between p and P may on these lines be inferred to exist, it is difficult to settle exactly what it is*. It seems doubtful that it would be such as to give the exact equality between B1 and ß, which experiment suggests unless the mutual energy between each pair and the rest of the group is negligible. (3) A theory which is to some extent intermediate between (1) and (2) has much to recommend it. We must infer from the preceding comparisons that the interionic forces must extend over considerable distances, as it is only The straightforward way to settle this point is to calculate p by a method in accordance with its definition on p. 357 and compare it with P. The calculation can be carried out strictly by exactly the same method as that by which P was originally determined (loc. cit.). It has been done, but unfortunately the numerical results in both cases can only be obtained in an approximate form which is not sufficiently accurate to determine definitely what is the difference between them. in this way that a satisfactory explanation of the failure of the mass action law can be got. On the other hand, the idea of the molecular pressure of a pair of associated ions seems also necessary to obtain an accurate agreement with the experimental equality of the freezing-point and conductivity variations. Both conditions will be satisfied if the law of interionic force be such as practically to confine the electrical attraction to pairs of nearest ions. Now, the view that in an electrolyte each + ion attracts every ion and repels every + ion is undoubtedly a highly artificial one. How artificial it is is made evident by observing that the mutual energy of an ion with others would have to be expressed as a sum of hundreds of terms before any close approximation to its value could be obtained. These represent the mutual energies with the nearest ion, the next nearest, the third nearest, &c., and form terms which partly cancel each other as the successive ions are + and It is unlikely that this state of things represents a physical reality. Indeed, the assumption on which it is based, that the action of the water molecules can be simulated by that of a continuous medium of S.I.C. the same as that of water in mass, is hardly likely to be true. It is probably a good deal nearer to the truth to imagine surrounding each ion a number of polarized water molecules which tend to form chains linking together pairs of temporarily nearest oppositely charged ions. Such an action would not be the same as that of a uniform medium; it would be more analogous to the action of iron filings in forming chains between two magnetic poles. The general effect would be to increase the attraction between an ion and the nearest one to it of unlike sign at the expense of the attraction of more distant ones, which latter might well be negligible in consequence. The view of the constitution of an electrolyte which is thus attained will satisfy both the requirements mentioned above, which are essential to a satisfactory theory. On it we may imagine all the ions divided into pairs formed of ions which are temporarily nearest together, the individuals of each pair undergoing continual change. Between the ions of each pair electrical force exists, which in many ways is similar to a chemical bond, but is different in others. Thus, with chemical association, an ion is either free or combined, it cannot be both together, but here it possesses simultaneously characteristics of both conditions. The ions in each pair, for instance, will be separated spatially from each other nearly as widely as if they were quite free; they are free (e. g. to carry current or exert ionic pressure) in a fraction of the cases in which the pair is observed, while they act as combined (exert a molecular pressure) in the remainder. It seems to me that it is a theory on lines similar to these which will ultimately succeed in reconciling all the difficulties connected with strong electrolytes. SUMMARY. (1) A critical discussion of the way in which the law of mass action fails for strong electrolytes leads to the conclusion that the reduction in the molecular conductivity with increasing concentration must be ascribed mainly to a reduction in the mobilities of the ions, and not to a reduction in their number by association into molecules. (2) A theoretical investigation of the effect of interionic force shows that identical variations with the concentration will be produced in the conductivity and in the osmotic pressure of the "free" ions (as defined on p. 357). (3) The application of this result to strong electrolytes shows that the variation in the conductivity and the freezingpoint can be best explained by a modification in the view we take of what constitutes association. According to this ions in strong electrolytes are not associated into molecules; they are neither completely associated nor completely free, but pairs of ions which are temporarily nearest together, in consequence of the electric forces between them, will, in a fraction of cases, act as if bound together, and in the remaining cases as if free. The University, Sheffield, XL. Bessel Functions of Equal Order and Argument. By G. N. WATSON, M.A., D.Sc., Assistant Professor of Pure Mathematics at University College, London *. 1. A PPROXIMATE formulæ for the Bessel function and its derivate, J„(n) and Jn'(n), (when n is large) have been discussed in numerous papers during the last few years t; several of these papers have appeared in this Magazine. *Communicated by the Author. + Debye, Math. Ann. lxvii. pp. 535-558 (1909). Rayleigh, Phil. Mag. Dec. 1910. Nicholson, Phil. Mag. Dec. 1907, Aug. 1908, Feb. 1910. Watson, Proc. London Math. Soc. (2) xvi. pp. 150-174 (1917); Proc. Camb. Phil. Soc. xix. pp. 42-48 (1917). Various numerical results have also been given by Airey in a series of recent papers in the Phil. Mag. which does not occur in many of the physical problems in which the other Bessel functions present themselves, appears to play a prominent part in connexion with various series arising in the theory of Electromagnetic Radiation, and consequently Professor Schott has asked me to determine whether there is any approximate formula analogous to the results This note, in which I prove the remarkably simple result that is the outcome of his inquiry. A closer approximation is given in § 4, but it involves the gamma function of 1/3 ; this more precise result is * (J£wz)dzeea; alsn!T{རྙ) 3n In order to obtain this approximate formula I propose to employ not the elementary methods which I have used elsewhere in connexion with Jn(n) and J'(n), but the methods which depend on the contour integrals of Debye; the latter methods yield the desired result with a much smaller expenditure of labour. 2. We take the well-known contour integral ∞), and on (in which the contour starts from -∞, encircles the origin once counter-clockwise, and then returns to integrating under the integral sign we get |