THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 255 



N (tan &*> = (B + B, -|- B/ + B/) v/T, 



+t (B - B, + B 2 * - B,) yr, (16), 



B, B ;! = /SI + 7, B,, B 2 = 3v/51 + 19, N = 192 ^3 (17). 



27. These expressions for I (w) are applicable immediately to the construction of a 

 series of quasi-algebraical Poinsot herpolhodes, in continuation of the one invented 

 by HALPHEN (' F.E.,' 2, p. 279). 



Making a digression on the motion of a rigid body about a fixed point under no 

 forces, as illustrated by the motion of a body about its centre of gravity when tossed 

 in the air, POINSOT'S polhode and herpolhode are obtained by rolling the momental 

 ellipsoid on the invariable plane. 



The equation of the momental ellipsoid can be written 



Ax 2 + B*/ 2 + Cz> = D8* (1), 



where 8 denotes the, distance of the invariable plane from the centre of the momental 

 ellipsoid ; and then the polhode will be the intersection with the coaxial quadric 



AV + B-//- + G-z- = D'-S 3 (2), 



and the direction cosines of the invariable line will be 



A B?/ G ,. 



DS' T)8' DS 



Denoting by p, nr the polar co-ordinates for the herpolhode in the invariable plane, 



* 2 + r + z 3 = p 2 + s 2 (4), 



and by solution of these equations (l), (2), (4), 



= p- p,~ suppose, &c. (5), 



and then 



where 



R = 4 (p* - p~) (p? - p 3 ) (p" - p 2 ) (7). 



Denoting the component angular velocities about the principal axes by p, q, r, and 

 about the invariable line by h, 



__? - v - h - ri (8), 



"" ^ " 7 \ /' 



x y z o fc 



