( 733 ) 



or c^ sin t] {■= {/ a^ . ip^) put equal to u : 



jttj^ a^ 11^ = F^ (m), in which we can easily A^erify that F^ (u) has two 

 real roots between — c^ and -j- c^ (the two other roots are real 



outside those limits or imaginary according to 363"— &i' being ^ than 0); 



those two roots indicate the limits between which ?] swings to and 

 fro according to a course indicated by elliptic functions. For the 

 case 2^3" ]> b^^ for instance, thus for four real roots i^j <] u^ <^ u^ <^ u^, 

 that course becomes : 



u^ _|_ LJ_^_J ^ where sn = sn \^t \/ ' ' 1 3 v. 3 ^ 



l"*'3 3 "'S 



and 



Farther more : 





Un^Un 



p"'q 



^1 , 8^8 3^3 • 3^3 COS F 



F = — yp^ ^1 1 



3«3 3<^3 3^3 



where the second member is a rational function of (j\ (ipi can be 

 rationally expressed in (f^ according to (I), ^(f^^ according to (II), 

 3<;p8 v^Vz cos F according to (IV)), so that f^ too can be expressed in t 

 by elliptic functions and by that the entire "cone in the solid" ; and 

 further according to the above method also "the cone in space". 



The following special cases can very easily be traced to the end. 

 1*^ . The four squares of inertia are equal two by two. This case 

 is obtained by putting A^:=^ A^z=i 0. 

 Then 



a,=R' — A,' b, = RA, 



,a, =z R^ ,63 = 0. 



And the equations of motion pass into : 



ttj ^j rz: «J tpj =: 



2«S ^<PZ — 3«8 Az = 



F^z=—tp, F.p — —if), 



3*^3 3^8 



F = — (r/i — tf?i) 



3«8 



from which we directly read, ^j, 3^3, ipj and jifjj remaining constant, 

 that the "cone in the solid" for S,- and for Si is a cone of revolution 

 with the X axis as axis of revolution. Farthermore the moment 



