which is evidently the same as would be obtained by supposing the vibrations to take place in vacuo, under the influence of the given exterior force, provided the density of the vibrating body were increased from 2-a'b'c -% We thus perceive, that besides the retardation caused by the loss of weight which the vibrating body sustains in a fluid, there is a farther retardation dne to the action of the fluid itself; and this last is precisely the same as would be produced by augmenting the density of the body in the proportion just assigned, the mo ring force remaining unaltered. When the body is spherical, we have a = b' «= c, and the proportion immediately preceding becomes very simple, for it will then only be requisite to increase p, tLe density of the body, by |, or half the density of the fluid, in order to have the correction in question. The next case in point of simplicity is where «' = c'; for then Jta>bc-),a'b = 2),a< *** If a > V, or the body is an oblate spheroid vibrating in its equatorial plane, the last quantity properly depends on the circular arcs, and has for value («" ~ i'V {J ~arc (tan = JF=V))} - a*{+^ • If, on the contrary, a' < V, or the spheroid is oblong, the value of the same integral is 1,,,, ,,,_», V+W-a"), b' Another very simple case is where c' = V, for then the first of the quantities (12) becomes, if a > V, and if a < V, the same quantity becomes ■ p--.y |«c (—^^ -1{+aT/_-r By employing the first of the four expressions immediately preceding, we readily perceive that, when an oblate spheroid vibrates in its equatorial plane, the correction now under consideration will be effected by conceiving the density of the body augmented from Ja'A'-a^'arc {tan--jd-Jt-bV(a«-V) 2(a',-?,'y-Ja'=6'+flVarcjtan=-^-r5T U6V(«,,-i', & I \\a —" )) When b' is very small compared with d, or the spheroid is very flat, we must augment the density from pt to />, + — —, p nearly; and we thus see that the correction in question becomes les3 in proportion as the spheroid is more oblate. In what precedes, the excursions of the body of the pendulum are supposed very small compared with its dimensions. For if this were not the case, the terms of the second degree in the equation (1) would no longer be negligible, and therefore the foregoing results might thus cease to be correct Indeed, were we to attend to the term just mentioned, no advantage would even then be obtained; for the actual motion of the fluid where the vibrations are large will differ greatly from what would be assigned by the preceding method, although this method consists in satisfying all the equations of the fluid's 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. |