212 Mr Greenhill, On the Rotation of a [March 20, 



where 



a r abcdX „ a r ahcd ^ no r a?>cd\ 



A = ^p\ Q W^' B ^ p lw^)P } c=27rp J W^)P ; 



p denoting the density of the liquid, and 



P 2 = (a 2 + X)(b 2 +X)(c 2 +X). 



These two equations will determine if and £ 2 in terms of A, B, C, 

 a 2 , b 2 , c 2 ; and if thrown into the form 



Q v *-R£* = b z B-a*A, 



Q'7f-R¥ = c 2 C-a 2 A; 

 then 



B m sa-ty-y) + (*^v + x y rf _ (^± _ y K 

 g _ sjVM) + ^f +1 p_ (i^ _ x y a . 



/A \ fl 



and therefore 



, _ (b 2 B - g 2 ^) ff -(c 2 C- a 2 A) B 

 v ~ QE' - 0i2 



(i 2 B - a 2 ^l) #' - (crO- a 2 ^t) Q 



r 2 = 



QB' - Q'R 



Captain P. A. MacMahon, R.A., has performed the algebraical 

 reduction, which is very heavy, of these expressions, and he finds 

 that 



2 _ 2 a 2 b 2 (b 2 -c 2 )A-a 2 b 2 {±a 2 -W-c 2 )B+c 2 (a 2 -V)(*a* + b*-c 2 )C 

 V ~ ** " (c 2 + a 2 + /*) (6 2 - cy {3a 4 - (c 2 - a 2 ) (a 2 - 6 2 )} 



_ 2 _ c V(6 2 -c 2 )^-6 2 (c 2 -a 2 )(4a 2 -6 2 +c 2 )5-c 2 a 2 (4a 2 -6 2 -3c 2 )a _ 

 ? " ^ (a 2 + b* + fi) (6 2 - c 2 ) 2 { 3a 4 - (c 2 - a 2 ) (a 2 - & 2 ) } 



and substituting for A, B, and C their values 



(a 2 -b 2 )j [(c 2 -4a 2 )\ 2 -{3a 4 -(c 2 -a 2 )(a 2 + 6 2 )}\]-p3- 

 2^p = fJ ' ' (c 2 + a 2 + JO (6 s - c a ) [Ho 3 - (c 2 - a 2 ) (a 2 - 6' J j } 



