16 Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING 



— (w sin 0) = K sin + X sin ff = Gr* cos sin + {Im sin 0) . r'* cos 6' sin 0', (5) 



o) sin . Q = Xcosfl' = (Aw sin d) .r'^cos^ff, (6) 



where Q is the angular velocity of OM {i. e. of the plane HOI) about OH. 



Also we have —r- = G (7) 



at 



Let p, p denote the perpendiculars from O on the tangent planes to the central ellipsoid 

 at /, L respectively, then p = r cos 9, p = r cos Q' . 



Equation (4) becomes by (1) — (Ap^) = Gp% whence by (7), jo is constant. This shews 



that the tangent plane at / to the central ellipsoid is fixed, and that the central ellipsoid 

 therefore rolls on it as a fixed^ plane. 

 Also by (4) and (5) 



l(!^ = iff!L!!!L^U^' = Ay^tan0.tan0', (8) 



dt dt \w cos 61 ft) cos d 



and from (6) Q=V' (9) 



31. Now r, /, r" being conjugate radii of the central ellipsoid, there exist three 

 relations between them and the conjugate axes ; these are, (putting p sec Q, p sec for r, r 

 respectively and denoting the angle lOL by y) 



p" sec' d + p^ sec' Q' + r"' °=l + ;g + ?i = -^' suppose, 



y/'2 + y V" + p'i)'' sec' Q sec' e' . sin' X = ;b^'^^"*"2b"'^' ^"PP°^^' 



j,2p'2/'2 ^ ___ = G, suppose, 



and by reason of the rectangularity of the planes lOM, LOM, we have i 



cos j^ = sin sin Q'. 

 Eliminating r" and ^, we obtain 



C 



p' sec' 6 + /' sec' 9' + -^-^j = £, 



G f-^ + 4) + py'(sec' + sec^ 9'-l)^F. 

 From these eliminating sec' 9', we obtain 



which, (remembering what E, F, G denote, and putting a, /3, 7 for the three quantities 



1 1 1 .• 1 X 



, , , , ,1 ^ , 1 , 1 — — - respectively) 



Ap" Bp^' Cp" ^ ^' 



is equivalent to 



p'2 = p2(i + a/37 cot' 9); 



