26 Mr Basset, On the Stability <>f [Nov. 28, 



Now employing the notation of Hydy'o dynamics, §§ 363 — 367, 



'd\ 



so that 



bh^ 2 _ , r"(^\ 



l = \M{a'+b'), W = \M7rpahc\ 



O O J Q 



V = 





whence if the suffixes denote ditferentiation 

 V -- o/i-'g MQ 



'^~ M (a^ + Iff ^ba [da a dc ) oa' 



y ^ 20h' ah M/dQ_cdQ 

 ''^~ Mia'+by^ ba\db adc 



Now h = ^M(a^ + ¥)(o, 



and on substitution in the first two, we obtain the usual conditions 

 of steady motion of a Jacobi's ellipsoid, which includes as a par- 

 ticular case Maclaurin's spheroid. Accordingly when a = b, we 

 have 



Substituting in the last two equations, and omitting the factor 

 ^M, we obtain when a = b, 



1 fdQ _ c ^^ 

 a \da a dc , 



' aa — ' bb ~ 



"^ a\db adcJ^a'' 



The value of SV may accordingly be written 



8V=i(Vaa+ Va,)(Ba + 8by + i(Vaa- F„,) (8a - 86)^ 



The value of Vaa + Vab, as shown in my book, is always positive ; 

 accordingly if the displacement were spheroidal, so that ha = Sb, 

 the steady motion would be stable ; but if the displacement is 

 ellipsoidal, the steady motion will be unstable unless Vaa — 1^«6 is 

 positive. The condition of stability is therefore that 



