Supposing now s and i to be real quantities, and going back to (38), to investigate the convergency of the series for p (*) and Hence, when μ(1-e), where e is an infinitely small positive quantity, d αμ K> 8 or K < 8. Hence if is, the quantities within the brackets under in (38) vanish when μ = 1; and as they vary con tinuously, and within finite limits, when μ is continuously Acquisition increased from 1 to +1, it follows that Pi (8) (8) (s) of roots with vanishes one time rise of order. and one time more than does (8) P. Now (8) more than does looking to (38"), and supposing (as we clearly may without loss. of generality) that s is positive, we see that every term of p is positive if i<8+1. Hence if i is any quantity between The rootless 8 and 8+1, vito P vanishes when μ±1, and is finite and order. positive for every intermediate value of (8) μ. (8) Hence and from the second formula of (38), q vanishes just once as μ is increased continuously from 1 to +1: thence and from the first of (38), p, vanishes twice: hence and from (8) (8) (+1 the second again, qvanishes thrice, and so on. Again, as the (8) form of lowest coefficient of every term of the series (38") for q is positive The other (a) (s) form of lowest one root, when i < 8 + 1, this is the case for q, and therefore this func- order has tion vanishes only for μ = 0, as μ is increased from 1 to + 1. zero. Hence P vanishes twice; and, then, continuing alternate applications of the second and first of (38"), we see that 94+1 i+1 vanishes thrice, P12 (s) four times, and so on. Thus, putting all (s) together, we see that has j or j+1 roots, and (8) i+j-1 (8) P has j+1 or j roots, according as j is odd or even; j being any integer and i, as defined above, any quantity between 8 and 8 +1. In other words, the number of roots of p, is the even Census of number next above i-s; and the number of roots of q is the seral harodd number next above i-s. Farther, from (38vil) we see that any order. the roots of p lie in order between those of of between those of p, [Compare § (p) below.] These properties of the p and q functions are of paramount importance, not only in the theory of the development of arbitrary functions by aid of them, but in the physical applications of the fractional harmonic analysis. In each case of physical application they belong to the foundation of the theory of the simple and nodal modes of the action investigated. They afford the principles for the determination of values of i-s, Electric induction, motion of water, etc., in space between two coaxal cones. Electric induction, motion of water, etc., in space between spherical surface and two planes meeting in a diameter. values of 0. This is an analytical problem of high interest in connexion with these extensions of spherical harmonic analysis: it is essentially involved in the physical application referred to above where the spaces concerned are bounded partly by coaxal cones. When the boundary is completed by the intercepted portions of two concentric spherical surfaces, functions of the class described in (0) below also enter into the solution. When prepared to take advantage of physical applications we shall return to the subject; but it is necessary at present to restrict ourselves to these few observations. (m) If, in physical problems such as those already referred to, the space considered is bounded by two planes meeting, at any angle, in a diameter, and the portion of spherical surface 8 in the angle between them (the case of s<1, that is to say, the case of angle exceeding two right angles, not being excluded) the harmonics required are all of fractional degrees, but each a finite algebraic function of the co-ordinates §, 7, ≈ if 8 is any incommensurable number. Thus, for instance, if the problem be to find the internal temperature at any point of a solid of the shape in question, when each point of the curved portion of its surface is maintained permanently at any arbitrarily given temperature, and its plane sides at one constant temperature, the forms and the degrees of the harmonics referred to are as follows: These harmonics are expressed, by various formulæ (36)...(40), etc., in terms of real co-ordinates, in what precedes. () It is worthy of remark that these, and every other spherical harmonic, of whatever degree, integral, real but fractional, or imaginary, are derivable by a general form of process, which in- Harmonic cludes differentiation as a particular case. Thus if () functions of all degrees denotes derived from that of degree - 1 an operation which, when 8 is an integer, constitutes taking the by generalsth differential coefficient, we have clearly where P, denotes a function of 8, which, when s is a real integer, becomes The investigation of this generalized differentiation presents difficulties which are confined to the evaluation of P,, and which have formed the subject of highly interesting mathematical investigations by Liouville, Gregory, Kelland, and others. 1 ized differentiation. obtained alized in If we set aside the factor P., and satisfy ourselves with deter- Expansions of partial minations of forms of spherical harmonics, we have only to apply harmonics Leibnitz's and other obvious formulæ for differentiation with any by common formulæ, fractional or imaginary number as index, to see that the equiva- with generlent expressions above given for a complete spherical harmonic dices. of any degree, are derivable from by the process of generalized differentiation now indicated, so as to include every possible partial harmonic, of whatever degree, whether integral, or fractional and real, or imaginary. But, as stated above, those expressions may be used, in the manner explained, for partial harmonics, whether finite algebraic functions of έ, ŋ, z, or transcendents expressed by converging infinite series; quite irrespectively of the manner of derivation now remarked. degrees use arbitrary of rare to be (0) To illustrate the use of spherical harmonics of imaginary Imaginary degrees, the problem regarding the conduction of heat specified ful when above may be varied thus:- Let the solid be bounded by two functions concentric spherical surfaces, of radii a and a', and by two expressed. cones or planes, and let every point of each of these flat or conical sides be maintained with any arbitrarily given distribution of temperature, and the whole spherical portion of the boundary at one constant temperature. Harmonics will enter into the solution, of degree Derivation of any harmonic from that of degree -1 indicates the character and number of its nodes. where j denotes any integer. [Compare § (d') below.] Converging series for these and the others required for the solution are included in our general formulas (36)...(40), etc. (p) The method of finding complete spherical harmonics by the differentiation of 1, investigated above, has this great advantage, that it shows immediately very important properties which they possess with reference to the values of the variables for which 1 they vanish. Thus, inasmuch as and all its differential coeffi cients vanish for x=±∞, and for y=±∞, and for ≈=±∞, it follows that (8) Expression of an arbitrary func tion in a series of surface harmonics. Preliminary proposition. [Compare with the investigation of the roots of p and q in § (1) above.] The reader who is not familiar with Fourier's theory of equations will have no difficulty in verifying for himself the present application of the principles developed in that admirable work. Its interpretation for fractional or imaginary values of j, k, l is wonderfully interesting, and of obvious value for the physical applications of partial harmonics. Thus it appears that spherical harmonics of large real degrees, integral or fractional, or of imaginary degrees with large real parts (a + B-1, with a large), belong to the general class, to which Sir William R. Hamilton has applied the designation "Fluctuating Functions." This property is essentially involved in their capacity for expressing arbitrary functions, to the demonstration of which for the case of complete harmonics we now proceed, in conclusion. (r) Let C be the centre and a the radius of a spherical Let P be any external or distance from C. Let do Then, ƒƒ denoting an integration extended over S, it is easily proved that |