Liquid Spheroid and the Genesis of the Moon. 259 



so 



Hence the equations of motion as given by (13) become 



+ 



{a-hf^ (a + h) 



t2 



(Ti:^ + (T+^^-^*=^w&B-^, > 



(14a*) 



-ic=|7pcC- -, 



where 



T=i(a-6)^[f+ri|±^], 



We can show that t, / are constants, by forming the equa- 

 tions of conservation of moment of momentum and of vortex- 

 strength. 



The moment of momentum round the axis z is 



JJj{^(? + ^^^2)+/(?-^^2^)}^*'^^^^=const., 



or 



(c? + 6^)r+ Q ^""^ f/ = const. =t + t' 

 ^ -^ a'' + 6^ 



The surface-integral of vortex-strength over any surface 

 bounded by the principal section in the plane xy is propor- 

 tional to ^ab, so that 



fa6= const. =:i('/— t). 



Hence t and ■/ are both constants. 



5. We are now going to suppose that the motion given by 

 (14a) is a small oscillation about the state of steady motion, in 

 which fl = and ^is a given constant ?=0) = \/47r7p6. 



On eliminating a, the equations (14a) become 



"^^ 4. ^^_i A'_^Vi^rB7>2_C^2'|_'[7PKsav " 

 l^Haf^ (h + af A h)-^ h ^^^ -^O- J J^say, 



* These equations were given by Riemann {Ges. Werhe, p. 183). They 

 are here deduced by a method similar to that employed by Basset for 

 the general case. See Proc. Lond. Math. Soc. vol. xvii. p. 255 ; or the 

 second volume of his * Hydrodynamics,' chap, xv . 



S2 



.(15) 



