Now, all over the surface, e=0. Hence unless the functions denoted by ƒ, and ƒ, are identical*, or only differ by a numerical factor, we must have Now e is proportional to the thickness of the shell at any point. Calling this thickness de, we have therefore Hence, adding together the results obtained by integrating successively over a continuous series of such surfaces, we get 2 V1, V, now representing solid ellipsoidal harmonics, and the integration extending throughout the whole space comprised within the ellipsoid. This may be shewn more rigorously by integrating through the space bounded by two confocal ellipsoids, defined by the values λ and μ of e. We then get, as in the text, [fev, v. (F; (w) - Film) f() + (^)}} as=0; Now the factor within {} cannot vanish for all values of X and μ, unless the functions devoted by f1 and f2 be identical, or only differ by a numerical factor. 15. It will be well to transform the expression JJe vv, as 1 2 to its equivalent, in terms of vu, v'. For this purpose we observe that if ds, ds' be elements of the two lines of curvature through any point of the ellipsoid, dS= ds ds'. Now, ds2 is the value of da2 + dy2+ dz2 when e and v' are constant, + z2 -1, we get c2 + w (v — w) (v' — w) = (c2 + ∞)2 - (a2+w) (b2 + w) (c2 + w) (v' - w) (e-w) (e — w) (v — w) (v′ – w) + + y2 جمع = (vv) (e-v) (a2 + v)2 + (b3 + v)* + (c2 + v)2 = (a2 + v) (b2 + v) (ca + v) 16 (a2+v) (b2+v) (c2+v) (a2+v′) (b2+v′) (c2+v′) 1 d S2 (a2 + €) (b2 + €) (c2+e) ' writing e for w in the expression above; 1 (a2+e) (b2+e) (c2+e) (v′ —v)2 16 (a2+v) (b2+v) (c2+v) (a2+v′) (b2+v′) (c2+v′) It has been shewn that, integrating all over the surface, the limits of u are c2 and -62, those of v', - b* and — a2. Hence, V1, V2, denoting two different ellipsoidal har monics lent c2 1 2 V1V, (v' — v) dvdv -b2 −a2 {(a2+v) (b2+v) (c2+v) (a2+v′) (b2+v′) (c2+v')}* = = 0. The value of the expression Vardyde, or its equiva -b2 ) − a2 {(a2+v) (b2+v) (c2+v) (a2+v′) (b2+v') (c2+v') }1 ' in any particular case, is most conveniently obtained by expressing Vas a function of x, y, z. F. H. 9 16. Before proceeding further with the discussion of ellipsoidal harmonics in general, we will consider the special case in which the ellipsoid is one of revolution. We must enquire what modification this will introduce in the quantities which we have denoted by a, ẞ, y, viz. We will first suppose the axis of revolution to be the greatest axis of the ellipsoid, which is equivalent to supposing 2=c2. To transform a and y, put a2+y=03, a2+e=n2, a2+v=w2; we then obtain To transform ẞ, we must proceed as follows. Put ccos-b2 sin'w, v=-c2 cos2 -b sin3, =- we then get generally b2 + y = (b2 — c3) cos2w, c2+y= (c2 — b2) sin3 ~ ; d↓ = 2 (c2 — b2) cos a sin a da; Also, en2-a2, v' = w2- a2, v=-b', and our differential equation becomes This equation may be satisfied in the following ways. First, in a manner altogether independent of o, by supposing V to be the product of a function of ŋ and the same function of w, this function, which we will for the present denote by ƒ (n) or ƒ (w), being determined by the equation |