( 731 ) 



If we put R"" — Afj^ = a^j and RAp -\- Aq A, = bp, and if we 

 represent the vector a^ ip^ i + a^ cp^j -\- a^ (p^ k, by (a) (f, we can write 

 the six equations of motion : 



(a) ip= V.{b)xp.(p^ 2Rii — 2Av 

 {a)rp= V. (b) <p.xp i- 2Rv — 2A^i. 

 For absence of external forces: 



(a) (f z= V . {b) \\i . (fi j 



(h) 



(a) ip = F . (6) y . ip I 



In this form we can directly read : 



l'*^ If in the initial position (p = tf', then <p remains equal to if% 

 i. e. if the initial rotation of S^ is a rotation // to a principal space 

 of inertia, then the motion remains a rotation // to that space. The 

 equations of motion for that case can be reduced to a system to be 

 treated as the well known Euler motion in S^ when the forces are 

 missing. 



2'^'^. For unequal ^'s "invariable rotating" is only possible under 

 the following two conditions which are each in itself sufficient: 



a. for (f and tp both directed along one and the same axis of 

 coordinate {X-, Y- or ^-axis of the representing spaces) i. e. for a 

 double rotation of S^ about a pair of principal phmes of inertia; 



b. for (p = or xp = 0, i. e. for an equiajigular double rotation 

 of S,. 



It has been pointed out by Kotter (see "Berliner Berichte" 

 1891, p. 47), how a system of equations analogous with what was 

 given, can be integrated. (The problem treated there is the motion of 

 a solid in a liquid). According to the method followed by him the 

 six components of rotation can be expressed explicitly by hyper- 

 elliptic functions in the time. If however we have y^, y-^, «/g, if>,, tf'j, if'j 

 expressed in the time, we have the "cones in the solid" for Sr and 

 Si. To deduce from these the "cones in space" we set about as 

 follows. We notice that the moment of motion to the right i?y + (yl) ip 

 in S^ remains invariable "in space" (in Si that vector changes of 

 course its direction, but there Rip -\- {A) p remains invariable) ; calling 

 the two spheric coordinates ("polar distance" and "length") of <ƒ; with 

 respect to Rp + (A) jp during the motion of .p in space ^h and / and 

 remarking that each element of the "cone in space" at the moment 

 of contact coincides with the corresponding element of the "cone in 

 the solid", we can express i^, x^ and x in the time, with which the 



