KIGID BODY ABOUT A FIXED POINT. 275 



a rotation about /; second, a rotation <p about the new position of k; third, a 

 rotation r about the final position of what was originally j. The connection of 

 this process with that of § 10 is sufficiently obvious. 



e 



Here j* ( )i - ^ is the operator for 9, and converts k into 



OG x = n= (cos - +7 sin -J k (cos ^ -jam-^J 

 = ismd + k cos0 . 

 Next, the operator for <p is 



and converts j into 



OB = £ = ( cos ^ + sin | (i sin + & cos 0) ) «/ ( cos ^ — sin ^ (*' sin + k cos d) j 

 = — i sin (?» cos 6 + j cos <p + k sin <p sin . 

 Hence we have 



T (j) 



[T T "I / <p <f> \ / 0\ 



cos g + sin3( - i sin cos 6 + j cos cj> + k sin <£ sin 0) I f cos s + sms (* sin + & cos 0) W cos o + ./'sin s 1 



r t t .. „ . ,. . ,r\ i <j> 6 . . <t> . 6 . <b . ,.0 6\ 



= cos H + sin s( - i sin <£ cos + 3 cos <£ + k sin <£ sin 6) ( cos k cos h + * sin <r sin o + J cos b sins + k sin g cos h ) 



t + <2> . 6-T .0 . . x + , - x . <£ 

 = cos — o — cos 2 ■*" * sm — 9 — sm ^ + .? sm ~~ 2~~ cos ^ + k cos — -^ — sin s • 



As a verification, we have by § 11 



OA= qiq- x 



= (w* + v? - y 2 - z*)i + 2 (wz + xy)j + 2 (ass — wr/) & 



= j cos (0 + t) cos 2 2" - cos C — T ) sin " 2 I ' ' + cos T sin ^ 3 ' + | sin ^ ~ T ) sinS 2 ~ sin ^ + T ^ cos> 2 J * 

 = (cos cos t cos <£ - sin sin t) i + cos t sin <£ j + ( - sin cos t cos <£ - cos sin -r)k . 



The coefficients of i, j, k, in this are the usual expressions for three of the 

 direction-cosines. The other six may be obtained by the same process. 



To express the angular velocities about OA, OB, OC in terms of the three 

 angles 0, <p, r, we have at once 



— «! = 2S . iq~ x q_ 



= 2(xw — wx + yz — zy) 

 = — 8 cos r sin <p — <j> sin r . 



And the others can be found in a similar manner. 



