58 Mr Green on ike Vibrations of Pendulums 



dx ~ ^J a?bc a 3 bc dx' dy a? be dy' dz a z bc dz 



and, consequently, at the surface in question, where f= 0, 



*±- X ,. r°df fxx df d$jxy_df d$ = p Z df 

 dx~ ■ PJ tfbc + a ' 3 b'd dx* dy ~ a'W dy' dz d^b'd dz 



A •£* A -f* A -f 



These values, substituted in (6.), give, when we replace ^, jh, and -j- 

 with their values at the ellipsoidal surface, 



-*+»M+& < 8 -> 



00 



which may always be satisfied by a proper determination of one of the con- 

 stants A and yu, the other remaining entirely arbitrary. 



From what precedes, it is clear, that the equation (2.), and condition to 

 which the fluid is subject, may equally well be satisfied by making 



* = ( x ' + rf&) » " nd * = ( x " + *f2& * ■- 



oo 

 provided we determine the constant quantities therein contained by means 

 of the equations 



o = x + s f°V + % a ndo = \» + s f°4L+ *K. 



r J alfc^a'b'd ^ r J abc° ^ a'b'd 



00 00 



respectively. The same may likewise be said of the sum of the three values 

 of <p before given. However, in what follows, we shall consider the value 

 (3.) only, since, from the results thus obtained, similar ones relative to the 

 cases just enumerated may be found without the least difficulty. 



Instead now of supposing the solid at rest, let every part of the whole 

 system be animated with an additional common velocity — X in the direc- 

 tion of the co-ordinate x. Then, it is clear, that the equation (2.), and con- 

 dition to which the fluid is subject, will still remain satisfied. Moreover, if 

 x', y r , z' are now referred to three axes fixed in space, we shall have 



x" — x — IXdt, y=y r > z = z! 



