case, as in the last, we see that, since yz when substituted for V, satisfies the equation 'V=0, the potential at any internal point will be V.; while, substituting for y, z their 0 bc values in terms of elliptic co-ordinates we obtain for the potential at any external point dy († + b2) (y + c2) { (y+a3) (y +b2) (y+c2) } $ dy (y+63) (y + c2) { (y+a2) (y+b2) (y+c2)} $ ° 29. We will next consider the case in which the potential, at every point of the surface, varies as x2= V suppose. This case materially differs from the two just considered, for since a does not, when substituted for V, satisfy the equation 'V= 0, the potential at internal points cannot in general be proportional to x2. We have therefore first to investigate a function of x, y, z, or of e, u, v' which shall satisfy the equation 'V=0, shall not become infinite within the surface of the ellipsoid, and shall be equal to a2 on its surface. Now we know that, generally (b2 + w) (c2 + w) x2 + (c2+ w) (a2 + w) y2 + (a2 + w) (b2 +w) z2 (a2+w) (b2 +w) (c2 + w) = (e — w) (v — w) (v'′ — w). And, if 0,, 0, be the two values of w which satisfy the equation (b2+w) (c2+w)+(c2+w) (a2+w) + (a2+w) (b2+w)=0...(1), we see that and ▼2 (e — 01) (v − 0 ̧) (v′ — 0 ̧) = 0, ▼2 (e−02) (v-0) (v' — 0) = 0. And, by properly determining the coefficients A,, A1, A2, it is possible to make A ̧+A ̧(e−0 ̧) (v−0 ̧) (v' — 0 ̧)+A ̧ (e−0,) (v−0 ̧) (v' —0,)... (2) 1 when b'c2x2 + c2a3y2 + a2b3z2 — a2b2c2 = 0. Der M10-2 2 Hence, the expression (2) when A ̧, A1, A, are properly determined will satisfy all the necessary conditions for an internal potential, and will therefore be the potential for every internal point. Now, we have in general (b2 + 01) (c2 + 0 ̧) x2 + (c2 + 01)' (a2 + 01) y2 + (a2 + 01)2 (b2 + 01) z2 − (a2 +01) (b2 + 01) (c2 + 01) = (e — 01) (v — 01) (v′ — 0 ̧) 2 Ꮎ (b2 + 02) (c2 + 02) x2 + (c2 +02) (a2 +01⁄2) y2 + (a2 +01⁄2) (b2 +02) ≈3 − (a23 +02) (b2 +02) (c2 + 02) = (e − 02) (v — 01⁄2 (v′ – 02) and, over the surface Hence, the surface, 2 being any quantity whatever, we have, all over (b3 + D) (c2 + d) x2 + (c2 + D) (a2 + D) y2 + (a2 + D) (b2 + D) z2 (0-6)6 (e — 0,) (v — 02) (v′ —.02) − I (D − 0 ̧) (≈ — 04), and therefore, putting = — a2, a2 (a2 +02) (e – 01) (v — 0, ̧) (v′ – 0,) (e — 02) (v — 02) (v′ — 02) + a2 (a2 + 0 ̧) (a2 +02). Hence, the right-hand member of this equation possesses all the necessary properties of an internal potential. It satisfies the general differential equation of the second order, does not become infinite within the shell, and is proportional to 2 all over the surface. We observe, by equation (1), that (l2+w) (c2+w)+(c2+w) (a2+w)+(a2+w) (b2+w)=3(0,−w) (02—w) identically, and therefore, writing -a2 for w, and we therefore have, for the internal potential, V1 V。 ((e−0 ̧) (v−0 ̧) (v'—0 ̧) ̧ (e−0 ̧) (v−0 ̧) (v′ —0,). 1 30, (0-0) (a2+0,1) 1 + 2+1}. 02 (02 = 0 ) (a2 + 0 ) ) + 1}. 2 2 This is not admissible for external points, as it becomes infinite at an infinite distance. We must therefore substi († − 0 ̧)2 { (y + a2) (y+b2) (y+c2)}} with a similar substitution for e-0, thus giving, for the external potential, The distribution of density over the surface, corresponding to this distribution of potential, may be investigated by means of the formula or its equivalent in Art. 13 of this Chapter. We thus find that 30. The investigation just given, of the potential at an external point of a distribution of matter giving rise to a potential proportional to x2 all over the surface, has an interesting practical application. For the Earth may be regarded as an ellipsoid of equilibrium (not necessarily with two of its axes equal) under the action of the mutual gravitation of its parts and of the centrifugal force. If, then, V denote the potential of the Earth at any point on or without its surface, and the angular velocity of the Earth's rotation, we have, as the equation of its surface, regarded as a surface of equal pressure, 1 dy :. V + √ Q2 (x2 + y2) = a constant, II suppose. Hence, if a, b, c denote the semi-axes of the Earth, we have, for the determination of V, the following conditions: |