H8 KEPORT— 1870. 



where A and C are the moments round the x^ and z^ axes, z^ being the axis 



of revohition, 

 y the distance of the centre of gravity from the plane of x^ y^, 

 (ii) the constant angular velocity round the axis of revolution, 

 (I) the moment of the quantity of motion of all the points of the body rela- 

 tive to the vertical axis of z, 

 (Ji) a quantity introduced by the integration. 



Then if Kp a„ a^ are the three roots of the cubic equation, 



when — O3 is greater than unity, and a^, a., lie between —1 and +1, h the 



modulus of the elliptic functions employed in the solution = ' ^ , so that 

 '^"i-S 



k = , , z^, sin''a}n la = , sm^ am (M^-\-K)=siTr coamia„ = -^ ^, 



Vai — a3 «, — "1 " l + cti 



where 



J 0^1— /.-sin- ^ 



V-(l + n,i 



Voj— O3 



H(i(«i+«,) +K) H(K«,-«,)-K)=D, 

 e(M— iaj)=Ai, e(M-f iai)=B', 

 0(H-{a3-K)=A", eOe+w,+K)=B". 

 Then a, a', a ', /3, &,g. being the same nine direction cosines as before, 

 _J^ HX(B"' + A"^)-H^(m,+K)(E"+A'') 



,_ 1 W ia^jB" ^ - A" ^)— H^m, + K) (B'"- A' ^) 

 " ~2(D ' e'M ' 



„_ H/«,HOX+^) B'A"-A'B" 



1_ HVa^(B"^-A"'=) + ffOVt,+K)(B'^-A") 



'^~ 2/D' e^t 



1 HX(B"^-A"-) + H-(m,+K)(E"+A'-) 



