in Fluid Media. 61 



00 



J a 3 be 

 P t *>p t + 9 06 p (11.) 



2— a'b'c'fJJL 

 J a 6 oc 



o 



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 re- 

 tardation due 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 moving 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 requi- 

 site to increase p t the 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 a' — c', for then 



oc oo co 



rjL=r^L=c l r d ^ as.) 



J a 5 bc J a*b J « 4 

 o o y 



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 



b' 



K«-r ) -»{!-arc(ta I1=7c?f ^ FJ )} 



a* (a'2 _ J/*) 



If, on the contrary, a' ^l V , or the spheroid, is oblong, the value of the 

 same integral is 



a (* a ) l0 g V _ V(6 * _ a , 2) + # (ft « _ fl , 2) 



Another very simple case is where c' = b', for then the first of the quan- 

 tities (12.) becomes, if a'^- b' 



(a b } l0g a'-^-b'*) a'(a'*-b>*) 

 and if a! z^ b' ', the same quantity becomes 



