probably due the high temperature of the spark. This high temperature in turn is favourable to the production of fresh ions, and to this fact the comparatively steady state of the damping in circuits with large capacities may probably be attributed. In such a case the current is so great that a large number of ions are produced, the recombination of these produce a high temperature by means of which fresh ions are formed to take the place of those that have recombined, and the field is thus kept comparatively uniform. I wish to express my obligation and gratitude to Professor Rutherford for his kind assistance at every stage of the investigation. Macdonald Physics Laboratory, McGill University. Feb. 12th, 1901. V. A Further Note on van der Waals' Equation. IN a note T. collected mid various mathematicad Na previous note † I collected the various mathematical by considering the temperature a variable parameter in van der Waals' equation (the Isothermals); in the present note I propose to complete this work by doing the same for the curves of constant volume and the curves of constant pressure. We take the same equation as before, 3yx3-(y+80) x2+9x-3=0.... (a) We will first consider the family obtained by considering a as an arbitrary parameter. The family is a series of straight lines such that two consecutive members meet in the point Eliminating from these two [equations, we have as the equation of the envelope (y+80-9)3+81(y−40+3)2=0, 27 which has a cusp at the point (1, 1), the tangent to the cusp being y-40+3=0; cuts the axis of where 0=, and touches it where 0=0; cuts the axis of y where y=0, and touches it where y=-27; and goes off to infinity in the direction of y+80-9=0. A series of points on it are given in Table 1. *Communicated by the Author. The integral of this is the result of eliminating p between 16 p(3+2p) {845 (1-8p)2 31p sinh-1 8+P 1-8p + Cp 2 4225 √65 (1+p2) Any member of the family may be put in the form √1+p2 } The values of the denominators for different values of a are given in Table 1.; it will be noticed that the point where the line cuts the axis of y travels from zero to 91 as a goes from ∞ to 0, and from - to zero as a goes from 3 to -∞; while the point where the line cuts the axis of travels and x= When the line is 0=0, and when the line is y=0. In fig. 1 the lines of the family are drawn for various values of a (in some cases for the sake of clearness only the extremities are shown), together with the envelope and another curve discussed later. Again, let us take the equation 3yx3- (y+80) x2+9x-3=0,. and consider y as a variable parameter. (a) The equation now represents a family of curves each of which is of the 3rd degree and class, has one cusp and inflexion, no double points or bitangents, and is of deficiency zero. The cusp is in each case at infinity, the tangent being a=0; and 3+u the inflexion being where =1, 0= the tangent at the 4 inflexion being Each member has also an asymptote 1 below the curve when > crosses it when = , and lies above the curve when a< 1 The asymptotes all pass through the point (1, 0). A curve of the family (a) for which y is positive has a branch running to one end of X 8 1 y 3 8 =1, with an inflexion where =1, 1 39 crosses the axis of a where = and runs down to the negative end of the axis of 0 as asymptote; it has also a branch for which a and are always negative; the curve for which y=0 is similar, but has 0=0 as its asymptote; a curve for which y is negative has two branches as before, one for positive and one for negative values of x, but in this case each branch cuts the axis of a in a real point (where =+ 1 1 3 y these points lie further from or nearer to the origin than x=as y-27; the curve for which y=-27 touches the axis of x, where x, and the corresponding branch lies wholly below the axis of a (cf. fig. 3); the various curves of the family lie so close to each other between and x=0 that they cannot well be distinguished in a diagram. The parts of these curves lying between = and x-5-the parts x most interesting from a physical point of view-are given in fig. 3, together with another curve discussed later; fig. 2 gives a more complete tracing of two of the curves (y =5 and -2.5) together with the curve (B) (see below). 1 Tables II. and III. give a series of values of 0 for different values of y anda, from which the figures are drawn, putting (a) in the form The values of 3x-1 and 3(3x-1) are given in Table I. 8x2 The orthogonal trajectory of the family is dx 4x3 (3x-1) = −120x3+3(3x-1)2. de |