motion, and likewise the particular conditions to which it is subject It would be encroaching too much upon the Society's time to enter on the present occasion into an explanation of the cause of this apparent anomaly: it will be sufficient here to have made the remark, and, at the same time to observe, that when the extent of the vibrations is very small, as we have all along supposed, the preceding theory will give the proper correction to be applied to bodies vibrating in air, or other elastic fluid, since the error to which this theory leads cannot bear a much greater proportion to the correction before assigned, than the pendulum's greatest velocity does to that of sound. APPENDIX. Note to Art. 0, p. 86. Thb Important theorem of reciprocity, established in Art. 6, may be put in a clearer light by the following demonstration, which is due to Professor Maxwell. Let A, B be any two points on a closed conducting surface, and let a unit of positive electricity be placed at a point Q, within the surface, then a unit of negative electricity will be so distributed over the surface that there will be no electrical force outside the surface, and the potential outside it will be everywhere zero. The potential at any point P within the surface, due to the electricity on the surface, is a function of the positions of P, Q, and of the form of the surface. Denoting this by €rrw it is required to shew that Grm= or that the potential at P, due to the distribution on the surface caused by a unit of positive electricity at Q, is equal to the potential at Q, due to the distribution on the surface caused by a unit of positive electricity at P. Let Xbe any point outside the surface. The potential there is zero, hence *^i+^ro (i), where pA is the density and dSA the element of surface, at any point A of the surface, and the integration is extended over the whole surface. Also, by definition, $dSApA-L~G* (2). Now if we consider a unit of positive electricity placed at P, and if pg be the density on an element dSB at 8, we shall have, similarly, 2 ^sj»/lb 2£f + p Y"= °' for all points outside the surface, or on it, since the potential is zero on the surface. Let X be on the surface, say at A, this equation becomes lienre, substituting in equation (2) we get and as this is the same as we shall obtain for Gt>r\ the property is proved. Note to Art. 10, pp.-50, 51. The equation <f,{r) = * J proved on p. 51, may be expressed in words as follows. Let 0 be the centre of a sphere of radius a, and A, B, two points each of which is the electrical image of the other with respect to the sphere (i.e. let 0, A. B be in the same straight line, and OA. OB =a'}, then, if electricity be distributed in any manner over the surface of the sphere, the potential at A is to the potential at B as a is to OA or as OB is to a. For, if a point P move in such a manner that the ratio BP to AP is constant (=X suppose) it will describe a sphere, and if C, C be the points in which this sphere cuts AB, AC-gL, At,.", |