344 



Second Case Tlie Axis is restricted to a right Circula/r Cone. 



Let ( C) be the trace on the unit-sphere of 

 the axis of the cone (P) and (X) as before. 



Let (CX) the angular radius of cone 

 = a, {PC) = 7 angle PCX = ^ ; 



then 

 dt 



sma 



dt' 



Cos ^ = cos a COS 7 + sin a sin 7 cos ^. 



Equation (4) becomes, on substituting these 

 values, and dividing by sin ^7, 



\dtj [dtl 



sm a 



sm 7 



{m - to cos a cos 7) (cos I - COS lo) 



- sin 'a (cos 2|: - cos ^^0) • • • (6)* 



Confining ourselves to terms of the first order, and supposing, as before, 

 the axis started at relative rest, we have 



dt 



^ sm a 



2 Mto (cos t - COS ^q), 



sm 7 ^ ^ ^ ^ 



Hence it follows that the axis (X) does not go all round the cone, but 

 vibrates about that edge of the cone which makes the least angle with 

 the polar line, that edge for vfhich ^ = 0, The length of the equivalent 

 simple pendulum and the period of a double oscillation, when the vibra- 

 tions are small, may be found, as in the last case [which is, indeed, in- 

 cluded in this as a particular case] to be 



sm 7 

 sin a 



711 w 



^ moo sm a \ 



A sin 7 

 G sin a 



r. 



* Not long since, Professor Curtis, of Queen's College, Galway. published an interest- 

 ing paper on this subject. In his investigation of the question he has followed an entirely 

 different method from that here adopted. The origin of the present paper was an endea- 

 vour to trace out the cause of the difference between Professor Curtis' results and those 

 arrived at by Professor Price, of Oxford, in the chapter on the gyroscope, in the lately 

 published fourth volume of the Infinitesimal Calculus. 



The differential equations (5) and (6) for the motion of the axis, in the last two cases, 

 precisely agree with those given in Professor Curtis' pamphlet, and differ from the cor- 

 responding equations in Professor Price's work, — the reason being that the latter follows 

 M. Quet in his assumption, and writes the relative vis viva = Const. 



