RIGID BODY ABOUT A FIXED POINT. 301 



57. From (49) we have at once, by operating by S . y and integrating, 



BS.ya« = AQSya + C .... (52). 



Also, operating by S . Yya , 



B S . ya Vaa = AQS . 7 a« - (Vayf . . . (53), 



or 



B ( — Sya — SyaSaa) = AQS . yad + a 2 y 2 — S 2 «7 



— g- Sya + — g y'-bay, 



by (52). 



This may be written 



■d r o •• o / ™ 2S7a + C\~| A 2 Q 2 AfiC o Q2 

 B - Sya — Sya^- i} 2 £= J = -=^- Sya + -^ 7 2 -S 2 ay , 



which leads, by integration, to the ordinary expression for S-ya in terms of an 

 elliptic function. It is to be observed, however, that this quantity is not one 

 which the quaternion calculus directly points out as an object of research ; the 

 propriety of seeking a in the first place being clearly indicated. 



58. From the above equations all the ordinary results connected with this 

 problem may be at once deduced by any one who has a little skill in quaternion 

 analysis : but the determination of the quaternion which gives the position of the 

 body at any time does not appear, so far as I have yet examined the question, to 

 lead to any very simple expressions. 



If we could, generally, integrate equation (49), g would be at once given by 

 (47), and the determination of the motion would be reduced to comparative sim- 

 plicity. The equation for the direct determination of £ may be formed as 

 follows, but it is not so simple as that for a. 



From the equation 



Bi - (A - B)QV £a = V«7 , 



we have, by operating by V . s , the result 



BVsl - (A - B)Q(« S 2 - ,Q) = Qy- ccSyB . 



which gives 



_ BYei + (A -B)fi 2 s-Q y 

 " ~ (A - B)Qs 2 - Sye 



The condition 



a, = Vsa 



gives, by substituting this value of a, 



BW + (A - B)Q 2 I - - ^^-yg 2 ^^ (2(A - B)Q*i - Syi) 



= B(!e 2 - 6 Sss) - QVey . 

 VOL. XXV. PART II. 4 H 



