150 The Rev. Samuel Haughton's Account of 



dcp 



K'dt = 



Kdt^ 



(14) 

 d^ ^ ' 



^/{\ --^ sin-^Jr 



The motion of the principal axis of moments is, therefore, expressed by an el- 

 liptic function of the first kind. 



The motion of the axis of moments is determined by the magnitude of the 

 radius vector of the elHpsoid, which is the axis of the original couple impressed 

 upon the body; if this radius vector be greater than the mean axis of the ellip- 

 soid, the corresponding spherical conic will have the axis of x for its internal 

 axis ; and if the radius be less than the mean axis, the axis of 2 will be the internal 

 axis of the conic ; in no case will the mean axis be the internal axis of the 

 spherical conic. If the radius R be nearly equal to either the greatest or least 

 semi-axis, the expression (14) for the time may be integrated. Let R be nearly 

 equal to the greatest semi-axis. The first of the equations (14) belongs to 

 the interior circle, which is of small dimensions in the case supposed ; the 

 second equation expresses the vibratory motion of the projection, through a 

 small arc of tlie outer circle, which will have a radius much greater than the 

 inner circle ; we may, therefore, suppose the angle -f to be equal to its sine. 



Multiplying both sides of the equation by -^, we obtain 



C 

 ^' ^ dt ^ K'dt = ^ 



Hence 



-^ = s\n{K't + A). (15) 



If 7^0 denote the time of a complete oscillation or revolution of axis of moments 

 about the axis of x, and T^ the time of a revolution of the body round the 



