270 PROFESSOR TAIT ON THE ROTATION OF A 



originally coinciding with j; third, a, rotation (p about the final position of the 

 line at first coinciding with k. 



Let i, j, k be taken as the initial directions of the three vectors which at time t 

 terminate at A, B, C respectively. 



The rotation \^ about k has the operator 



h"( ) h v . 



This converts j into >/, where 



± _0 

 n — Tc 7 j k * =j cos 4 — i sin -^ . 



The body next rotates about n through an angle 0. This has the operator 



e _e 



n ( ) » 

 It converts k into 



e e 

 00 = Z, — jf k « * = (cos - + „ sin ^ ) k (cos - - n sin -J 



= A; cos 6 + sin (i cos -^ + y sin -vj/) . 

 The body now turns through the angle (p about £, the operator being 



* _0 



Hence 



<£ e 



/■ „ . 0\ /■ ^ -0\/ 7 ■ 0"\ 



= f cos -j, + £ sin n 1 ( cos o + '' Slu o ) I cos ~2~ + sm "2" / 



/ d) rf>\ r 9 0.0 .6 \// / \ . e . 0/ \-i 



= f cos I + I sin £ J cos s cos 9 + * cos g sm 2 + sln "2 cos 2 V cos V ~ l sm J + sul 2 sln 2 V l cos ^ + •? sm ^) J 



/■ <i> „ . \ r fl ^ ■ . 6 . 0- . . 6 , 6 01 



= f cos 7y + 5 sm -^ J cos o cos o - * sm s sm 9 + J sin g cos -k- + a; cos g sin -^- I 



<^> 6 xtr <^> «V <p <f> \ff 



= cos o cos g cos "2" + sm 2 sm „ sin -5- sin 6 cos - sin „ sin 5 cos -g- sm sm - sm s cos 5 sin y cos 



■ f <l> . rf> tf> . <p ■> 



+ 1 1 - cos 2 sm H sm "o" + sin o cos 5 cos -a sin 6 cos - sm g sm % cos -g- cos + sm „ cos « sin -5- sin sin 1 



. / 4> . 6 \1/ <t> 6 \1/ . 6 . „ (j> 6 xfr \ 



+ ^ I cos ^ sm g cos -A- + sin -r cos f> cos -5- sin sin0 - sin h sm ^ sm "2" cos - sm -„ cos « sm . g sin cos 01 



,/ . . 4> ^ d>.0.0... d>0 .1/ a 



+ A I cos ^ cos 2 sm -g- + sm ^ cos s cos -^ cos + sm h sm -s sin -g- sm sm \J/ + sm <j sm 9 cos -5- sin cos >^ ) 



<P +■& . . a - * .. e <t> - & ,.0 + 



= cos — 2 — cos g + 1 sm — n" — sin o + 3 cos n ' sm „ + k sm — n" — cos 2 



which is. of course, essentially unsymmetrical. 



