equator towards the spectator: and z towards the north pole. Then x = r. sin u. cos λ, y = r. sin u. sin λ, z=r.cos u. For changing our coordinates, we must put dv dV du dV dr dV dr + dx du dx dλ dx dr dx' and similar equations for y and z: where u, λ, and r, are supposed to be explicitly expressed (as was V) in terms Now W X sin λ- Ycos A. Also the force in the = direction of radius of the parallel passing through the point of observation = X cos λ + Ysin λ; from which, combined with the force Z, N= Zsin u-(X cos λ + Y sin λ) cos u, C = Z cos u + (X cos λ + Y sin λ) sin u. the same as the values found before. Every thing now depends on the function V: and or {(x − a)2 + (y − b)2 + (z − c)2} ̄1. For x, y, z, the ordinates of the experimental magnet, put their values (already used) r. sin u. cos λ, r sin u. sin λ, r cos u. And for a, b, c, the coordinates of a disturbing particle of magnetism, put similar coordinates, a=r. sin u. cos λ, br.. sin u. sin λ, c=r,. cos u (If the experimental magnet be on the earth's surface, and the disturbing magnetism be within the earth, r。 is always less than r.) The value of now becomes u.cos 1 ρ [r2 - 2rr {sin u. sin u..cos (λ — λ) + cos u. cos u} +r2] ̄1; which can be expanded in a converging series where T=1, and T1, T, &c., are functions only of u, u。, 0 2 and -λ. Put R for the earth's radius (the symbol r being still reserved for the radius at the place of observation, in order to preserve the generality which admits of differentiation with respect to r). Then V 2 =-ST1 r2 Sμ, &c. The general term will be 2 Now forming the values of N, W, C, and remarking that (as the integral with respect to du applies only to elements entirely independent of u, λ, and r,) the dif ferentiations with respect to u and λ can be performed under the integral sign, and the differentiation with respect to r will be entirely external to the integral sign; the general term of N or 1 dV r du will be Also it is to be remarked that T, is 1, and therefore ST. Su is 0 (because the total amounts of red and of blue magnetism are supposed to be equal), and therefore P is 0. And, if our needle be on the earth's surface, r R. Thus we obtain = C=+2P1 + 3P2+ &c. + (n + 1) P2+ &c. and T is the coefficient (in terms of u, u, λ, λ) of () 48. Incidental introduction of Laplace's Coefficients (not further used in this Treatise). If we differentiate twice the expression with respect to x, also with respect to y, and with 1 And since V: and since the application and limits of this integration do not depend on x, y, z, it Now, by the same principle which we have used in the last Article for V, and which we shall here use succes |