56 Mr. Spottiswoode — Equations of Rotation [1863. 



tlie expression becomes 



a:V(A + 0)4.ra 

 - C|^dr+ + (C?^ + H^)0 - CH0= 



= a,:VG + m, 



=V(A+0)+ra = vH +Tf^ = vG +m 



:VH +T?^ : V(B + 0) + T13 : VF +Tdr > • (1^) 



witb similar expressions for /3, /B^, /B^ ; y, y^,, y^, obtained by writing 0i, ; 

 02, respectively for 0, T. 



Returning to tbe equations of motion (1), and transforming by the 

 formulae 



(Aa+Ha,+Ga>\ = [~FK-a,^) + (B-C)a,a,+Ha,a-G«aJi.,^ 

 + (A/3+Hft+G/3j9', + [--F(ft^-/3/) + (B-C)Aft + HA/3-G/3A]^,^ 

 + (Ay+Ay, + Gy,)r', + [-F(y,^-y/) + (B-.C)y,y, + Hy,y-GyyJr,^ 



+ [-2F(fty,-^,y,) + (B-C)(fty, + ^,y,) 



' +H(Ay4-/3y,)~G(/3y,+Ay)]9r,r, 

 + [-2F(y,a,-^y,a,) + (B-C)(y,a,4-7,a,) 

 + H {y^a.^ ya,) - G (yor , + y^ft)] r^, 



+ [-2F(aA-aA) + (B-C)(aA+«A) 

 + H(a,/3 + a/3,) -G(a^,+a,^)];^,2r^ 



= [a/Ha + Ba, + Fa,) - a,(Ga + Fa, + Ca,)]_p,' 

 + [/3,(H/3 + Bft + F/3J-ft(G/3+Fft + CA)]^,= 



+ [y,(Hy + By +FyJ-y,(Gy + Fy, + CyJ]r,» 

 + [/3,(Hy + By, + FyJ-ft(Gy + Fy + Cy,) 



+ y,(H/3 + Bft + F/3J-y,(G/3 + F/3+C/3J]^,r, 

 + [y/Ha + Ba, + Fa,) - y,(Ga + Fa, + CaJ 



+ a,(Hy + By, + Fy,) - a,( Gy + Fy, + Cy J]r,p, 

 + [a,(H/3+B/3, + F/3,)-a,(G/3 + F/3, + C/3,) 



+AXHa+Ba, + Fa,)~./3;(Ga+F«,+Ca,)}i?,?„ 



whence the system 

 a : a, : a. 



: VG 



5=a,i?,+ft^, + y,r„ 



(16) 



we have 



