( 728 ) 



'OYi, ozr 



pXu OW, 



is no more equiangular to the right, but to the left, or if we suppose 



'OF/, OZ; 

 OX/, OW, 



to be also equiangular to the right, we must take as indicating vector 

 in Si that vector in the direction of which W would move together 

 with S^, not the one moving together with ^4 in the direction of TF. 

 We shall do the latter. The advantage of this choice will be evident 

 from what follow^s.' 



We have still to remark, that if only the position of Sj. and Si is 

 determined, the position of S^ ensues from it not in one, but in 

 two ways; for, a position of /S^ gives no other positions of S,- and 

 Si as its "opposite position" for which all vectors are reversed ; that 

 opposite position can be obtained by an arbitrary equiangular double 

 rotation over an angle jr; Sr and Si then rotate 2 n and are again 

 in their former position. 



But a continuous motion of S^ out of a given initial position is 

 unequivocally determined by the given continuous movements of 

 Sr and >S/ out of the corresponding initial positions. So we shall 

 have completely answered the question how a solid S^ moves under 

 the action of determined forces if we can point out how Sr and Si 

 move under those actions; in other words if we can point out "the 

 cones of rotation in the solid and in space". 



APPLICATION. The Euler motion in S,. 



The equations of motion for this have been given for the first 

 time by Frahm in the "Mathematische Annalen" Band 8, 1874 p. 35. 

 The system of- coordinates OX^ X^ X^ X^ in the solid we shall 

 choose in such a way that the products of inertia disappear. We 



shall suppose 



'OX,, OX, 



OX,, OX,^ 

 to be equiangular to the right. 



And we choose OXr ïr Zr and OXi Yi Zi in such a way that 

 OX,, OX,, 0X3, OX, 



OXr, OYr, OZr, OW 



is equiangular to the right and 



0X„ 0X„ 0X3, OX, 

 OXi, OYi, OZi, OW^ 



