62 Mr Green on the Vibration of Pendulum in Fluid Media. 



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 affected by conceiving 

 the density of the body augmented from 



£ a" b' - a'*b> arc (tan = _*_\ _ ft* V(a' 2 - b") 



P/ top /+ - j[± ^ s - - p 



2 (a' 2 — b' 2 ) § — -| a' 2 b' + a' 2 b' arc ( tan = /(q ,. 2 _ 6 , 2) ) + 6' 2 V a' 2 — b'- 



When b' is very small compared with a', or the spheroid is very flat, we 

 must augment the density 



from p to + — — - p nearly ; 



and we thus see that the correction in question becomes less in proportion 

 as the spheroid is more oblate. 



In what precedes, the excursions of the body of the pendulum are sup- 

 posed very small, compared with its dimensions. For, if this were not the 

 case, the term 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 vibra- 

 tions are large, will differ greatly from what would be assigned by the pre- 

 ceding 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 en- 

 ter 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. 



