\i 



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



If the moveable rectangular axes in § (15) be supposed fixed in the body and coincident 

 with the principal axes, we must substitute 



o)], («2> '^3 ^o** ^xi ^j,» ^zi and hi, Ag, A3, or Awi, Bw^, Ccos for A^, Aj,, A^, 

 and then we obtain three equations, of which the type is, either 



or A~=L +{B- C). «>,W3. 

 The latter is the well known form of Euler's equations. 



27. Instead of employing these equations, let us endeavour to solve our problem more 

 directly. Our object is to determine the motion of 01, the axis of rotation, both in the 

 body and in space, and the variation of w, the angular velocity about it. This may be 

 conceived to be due to an angular acceleration of definite intensity about a definite line ; and 

 this may be regarded as compounded of two similar accelerations, the one arising from the 

 acceleration of momentum produced by the couple G about its axis OG, the other being the 

 angular acceleration which would exist if no forces acted. Now the forces in the elementary 

 time dt produce the angular momentum Gdt about OG, and this momentum gives rise to a 

 corresponding angular velocity Kdt about an axis OK related to OG, just as 0/ is OH: thus 

 the angular acceleration *c due to the forces is determined as to direction and intensity. The 

 other component of the angular acceleration is in like manner due to a cori'esponding accele- 

 ration of momentum, which it is now necessary to determine. 



28. Regard any line OP fixed in the body and moving with it by reason of the velocity 

 10 about 01; and apply equation (C) of section I., putting A for m ; therefore 



dt 



= -hw . sin IH . sin HP . sin IHP, 



which determines the acceleration of momentum for any line OP. This acceleration will be zero, 

 if OP be in the plane HOI, and a maximum, if OP be perpendicular to HOI, when its value is 

 hot] sin HI: we may therefore regard the total acceleration* (/) due to the motion of the body 

 as being about the line OF, perpendicular to HOI, and equal to + hw sin HI, when OF is 

 taken on that side of HOI on which a positive rotation about OF would move OH towards 

 01. Now to this acceleration of momentum (/) about OF will correspond an acceleration of 

 angular velocity (\) about a line OL which is related to OF, just as 0/is to OH. 



29. To sum up our results, we have shewn that, if OH be the axis of angular momentum 

 (A) and 01 that radius of the central ellipsoid at whose extremity the normal is parallel to 

 OH, 01 is the axis of angular velocity (w) : if OG be the axis of the impressed couple (G), 

 and OK the radius for which the normal is parallel to OG, OK is the axis of angular accele- 



• This result is that which M. Poinsot states thus : " The 

 axis of the couple due to the centrifugal forces is perpendicular 

 at once to the axis of rotation and to that of the ' couple d'impul- 



sion.'" — M. Poinsot's "couple d'impulsion" is our angular 

 momentum. 



