i manner as Q, is derived from P. The general type of such expressions will be and this when multiplied by cos op or sin op, will give an expression satisfying the differential equation and which may be called the Tesseral Harmonic of the second kind, of the degree i and order σ. CHAPTER VI. ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 1. THE characteristic property of Spherical Harmonics is thus stated by Thomson and Tait (p. 400, Art. 537). "A spherical harmonic distribution of density on a spherical surface produces a similar and similarly placed spherical harmonic distribution of potential over every concentric spherical surface through space, external and internal." The object of the present chapter is to establish the existence of certain functions which possess an analogous property for an ellipsoid. They have been treated of by Lamé, in his Leçons sur les fonctions inverses des transcendantes et les fonctions isothermes, and were virtually introduced by Green, in his memoir On the Determination of the Exterior and Interior Attractions of Ellipsoids of Variable Densities, (Transactions of the Cambridge Philosophical Society, 1835). We shall consider them both as functions of the elliptic coordinates (as Lamé has done) and also as functions of the ordinary rectangular co-ordinates; and after investigating some of their more important general properties, shall proceed to a more detailed discussion of the forms which they assume, when the ellipsoid is a surface of revolution. 2. For this purpose, it will be necessary to transform the equation d' V dx2 d2 V ď2 V into its equivalent, when the elliptic co-ordinates e, v, v' are taken as independent variables. If a, b, c be the semiaxes of the ellipsoid, the two sets of independent variables are connected by the relations Thus a2 + e, b2 + €, c2+e are the squares on the semiaxes of the confocal ellipsoid passing through the point x, y, z. a2 + v, b2 + v, c2+v, the squares on the semiaxes of the confocal hyperboloid of one sheet. a2 + v', b2 + v', c2+v', the squares on the semiaxes of the confocal hyperboloid of two sheets. Thus, e is positive if the point x, y, z be external to the given ellipsoid, negative if it be internal. And, if a2 be the greatest, c2 the least, of the quantities a2, b2, c2, taken throughout a certain region of space, should be a minimum. In the memoir by Green, above referred to, this expression is transformed into its equivalent in terms of a new system of independent variables, and the methods of the Calculus of Variations are then applied to make the resulting expression a minimum. We shall adopt a direct mode of transformation, as follows: Suppose a, B, y to be three functions of x, y, z, such that ▼3α = 0, ▼2ß = 0, ▼3y=0.......................... .......(1), such also that the three families of surfaces represented by the equations a constant, B= constant, y = constant, intersect each other everywhere at right angles, i.e. such that a= d2 V dy2 d2 V and being similarly formed, we see that, when the dz2 three expressions are added together, the terms involving dV dV dV will disappear by the conditions (1), and those dev dev d2 V da' d' dy d2 V involving by the conditions (2). Hence dar {(de)2 + (dz) + (da)"} da2 d2V (dB 2 + + All these expressions satisfy the conditions (1), for a is the potential of a homogeneous ellipsoidal shell, of proper density, at an external point, and ẞ and y possess the same analytical properties. Again, a is independent of v and v', and is therefore constant when e is constant. Similarly ẞ is constant when v is constant, and y is constant when v' is constant. y satisfy the conditions (2). Hence a, ẞ, = dz (u* + e) (b* + €) (c* + e) { (dc)" (b2 (c2 dy dz + a2 + € c2 + € * for B. + · €)2 ' + But from the equations 4. ẞ is a purely imaginary quantity. We may, if we please, write√18 |