RIGID BODY ABOUT A FIXED POINT. 273 



Substituting the above values of £ and *i, multiplying out and arranging, we find 

 finally 



q = cos ^ cos - cos ~ — sin ~ sin - sin ^ 



2i 2» 2t 2* Ji L 



<p d . 4- . <b . 6 4A 

 - % ( cos ~ cos - sm | -f sin | sin - cos ^ j 



(' 



+ ^ ( cos \ sm 2 C0S f ~ Sm 2 C ° S 2 Sm I) 



7 / d> . d . 4/ . d> 6 4A 

 + ft ( cos ± sm -sin^ + sm - cos - cos ± j . 



The expressions for w 1? a> 2 , « 3 in terms of <£, 0, ^ and their differential co- 

 efficients are not very simple, and can scarcely be of any use. 

 We see by the equation of § 11 that 



— Wj = 2S . iq~ x q . 



If we put 



q — w + ix + jy + kz 



this gives 



— Wj = 2(xw — wx + y'z — zy) 



from which the required expression may be obtained. 



1 have not examined the question, but I fancy that to deduce the constituents 

 of the above value of q by means of spherical trigonometry would not be very 

 easy. 



13. To deduce expressions for the direction-cosines of a set of rectangular axes 

 in any position in terms of rational functions of three quantities only. 



Let a, /3, y be unit-vectors in the directions of these axes. Let q be, as in 

 § 7, the requisite quaternion operator for turning the co-ordinate axes into the 

 position of this rectangular system. Then 



q = w + xi + yj + zk 



where, as in § 8, we may write 



1 = to 2 + x 2 + y 2 + z 2 . 

 Then we have 



q— 1 = w — xi — yj — zk , 



and therefore 



a = qiq ~ 1 = {wi — x — yk -f zj) (w — xi — yj — zk) 



= (w 2 + x 2 — y 2 — z 2 ) i + 2 [wz + xy)j + 2 (xz — wy) k , 



where the coefficients of i, j, k are the direction-cosines of a as required. A simi- 

 lar process gives by inspection those of /3 and y. 



As given by Cayley, after Rodrigues, they have a slightly different and 



VOL. XXV. PART I. 4 A 



