RIGID BODY ABOUT A FIXED POINT. 271 



11. To find the usual equations connecting ^, 6, (f> with the angular velocities 

 about three rectangular axes fixed in the body. 



Having the value of q in last section in terms of the three angles, it may be 

 useful to employ it, in conjunction with equation (6) of § 8, partly as a verifi- 

 cation of that equation. Of course, this is an exceedingly roundabout process, and 

 does not in the least resemble the simple one which is immediately suggested 

 by quaternions. 



We have 



2q = sq— {u 1 OA+ « 2 OB + w 3 OC} q, 



whence 



2q~ 1 q= q-^^OA + o> 2 OB -\-^OQ}q , 



or 



2q = q(iu 1 + > 2 + £« 3 ). 



This breaks up into the four (equivalent to three independent) equations 



_d( <p + 4 6\ . <s — 4 . a £—4 -6 . <* + 4- & 



2 dt V 0S 2^ C ° S 2 ) = ~ Wl Sm 2^ Sm 2 ~ " 2 C ° S ~~2 S1U 2 ~ * 3 S1U 2^ C0S 2 

 2jl sm Q sm - J = «i cos J —^ — cos ^ — « 2 sm T — — cos - + w 3 cos ^— — - sin - 



dt V C ° S 2^ Sm 2 J = Wl Sm "2 C ° S 2 + * 2 C0S 2^ ° 0S 2 ~ " 3 Sm 2^ Sm 2 

 2 -t: ( sin ~ ,, cos - ) = — w, cos T „ T sm - + w 2 sin n ~ sm - + «, cos T 1 ■ cos - 



dt\ 2 2/~ 1 2 2 "2' 31 " 2 2 3 — o — 9 * 



From the second and third eliminate 4 — 4 , and we get by inspection 



6 8 



cos - . 8 = (wj sin <a + w 2 cos 0) cos - , 



or 



d = Wl sin <p + u 2 cos <*..... (8). 



Similarly, by eliminating 6 between the same two equations, 



sm - (<p — -4; = w 3 sm - + w 1 cos <4 cos - — u 2 sm <p cos - . 

 And from the first and last of the group of four 



tf.. ,. 8 .8 . 6 



cos - (<* + 4") = w 3 cos 7. — w x cos <* sin - + u 2 sm <p sm - . 



Z 2i Ji 2i 



These last two equations give 



4 + -4 cos 6 = w 3 (9). 



<p cos d + 4 =( — «i cos <p + u 2 sin <*) sin + w 3 cos 0. 

 From the last two we have 



4 sin 8 = — «! cos <p -f w 2 sin <p . . . . (10). 

 (8), (9), (10) are the forms in which the equations are usually given. 



