152 



The Rev. Samuel Haughton's Account of 



Fig. 3. 



IX. — To find the Position of the Body in Space at the End of any (jiven Time. 



First Method. 



The radius vector of the ellipsoid, which is perpendicular to the plane con- 

 taining the axes of principal moment and of rotation, always lies in the plane of 

 principal moment, and describes in that plane areas proportional to the time. 



Let OG, On be the axes of principal moment and 

 of rotation ; OR', Ofl', the axes of centrifugal couple 

 and of corresponding rotation ; the plane ilOfl' will 

 contain the two successive positions of the axis of 

 rotation. Let 01 be the position of the axis of ro- 

 tation at the end of the time U; then lu will be 

 equal to the angle described in the fixed plane by 

 the line OR'. Let R' and P' be the radius vector 

 and perpendicular corresponding to the centrifugal 

 couple and its axis of rotation. The following relations are evident from the 

 figure 



sin n'OI cos 0' _ ,___ „;„ ^,rM _ ^^^ </>' „,„ ^^^ _ sin #« 



to ~ sin n 01 sin 0cm 

 but from mechanical considerations, 

 us PR 



1 • ^//-vT cos <A . _,_ 



because sin ii Oi = — r-~ , sm nOI : 



sm 



sin 6 



IB PRm sin 0o< 



because, w = 



G 

 fxPR' 



Gw sin (j)ct 

 tj.PR' ■ 



Hence, by equating the geometrical and mechanical expressions, we obtain 



R'-cu = wPRct = — it. 



(18) 



The position of the body in space is thus reduced to quadratures ; but the 

 problem may be solved more readily in the following manner. 



Second Method. 

 The axis of principal moments, appearing to move in a direction opposite to 

 the rotation, describes in the body the cone whose equation has been given (11). 



