RIGID BODY ABOUT A FIXED POINT. 283 



It may suffice thus to have alluded to a possible mode of solution, which, 

 except for very simple values of ij, involves very great difficulties. The quaternion 

 solution, when n is of constant length and revolves uniformly in a right cone, will 

 be given later. 



28. If, on the other hand, we eliminate v, we have to integrate 



gf-i (g^yq) =2q , 



so that one integration theoretically suffices. But, in consequence of the present 

 imperfect development of the quaternion calculus, the only known method of 

 effecting this is to reduce the quaternion equation to a set of four ordinary differ- 

 ential equations of the first order. It may be interesting to form these equations. 

 Put 



q — w + ix + jy + kz , 



and 



y =. ia + jb + kc , 



then, by ordinary quaternion multiplication, we easily reduce the given equation 

 to the following set : — 



where 



and 



dt _dw _dx _dy _ dz 

 ~2 _ W _ X~T~ Z 



W = - a; a - y& - 2© or X = . y<& - zJ3 + w% 



X= w& + y® - z3S Y=-x€ . + z% + wB 



Y = w& + z% - x@ Z = x%-y% . + w<& 



Z= w<& + x& - y$L W = -x&-y&-z<K . 



% = -r- \ a (to 2 — x 2 — y 2 — z 2 ) + 2x (ax + by + cz) + 2w (bz — cy) I 

 B = ^ I b (w 2 — x 2 — y 2 — z 2 ) + 2y (ax + by + cz) + 2w (ex — az) I 

 — 7T c(w 2 — x 2 — y 2 — z 2 ) + 2z (ax + by + cz) + 2w (ay — bx) j 



(24), 







W, X, Y, Z are thus homogeneous functions of w, x, y, z of the third degree. 



Perhaps the simplest way of obtaining these equations is to translate the 

 group of § 21 into w, #, y, z at once— instead of using the equation from which 

 t, and n are eliminated. 



We thus see that 



n = m + /as + m . 



One obvious integral of these equations ought to be 



w 2 + x 2 + y 2 + z 2 = constant , 



