ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA. 323 



Another very simple case is where c = b f , for then the first 

 of the quantities (12) becomes, if a > b', 



and if a' < &', the same quantity becomes 

 2 PB&JF* {arc (tan 



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 con- 

 sideration will be effected by conceiving the density of the body 

 augmented from 



tan 



4J I \At IX I c\ ' I UC*AA / / /2 7 '2\ 



2 I Y f & o J 



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

 very flat, we must augment the density 



from p t to p t -t- 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 pen- 

 dulum 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 



212 



