302 PROFESSOR TAIT ON THE ROTATION OF A 



59. Processes very similar to these may be applied to the motions of the 

 Gyroscope and to Precession and Nutation. I confine myself at present to the 

 formation of the equation for the latter question, reserving for another com- 

 munication the details of the solutions of these three problems ; as they involve 

 some curious and delicate points of quaternion analysis. 



60. To form the equation for Precession and Nutation. Let a be the vector, 

 from the centre of inertia of the earth, to a particle m of its mass : and let f be the 

 vector of the disturbing body, whose mass is M. The vector-couple produced is 

 evidently 



T2 (? - «) 

 = M2 . m 



M2 



mYap 



/. 2S«g T*«\f 



V 1 + T 2 f + T 2 e ) 



MS. ^*(l -$» + *,). 



T 



no farther terms being necessary, since =- is always small in the actual cases 

 presented in nature. But, because a is measured from the centre of inertia, 



2 . ma = . 

 Also, as in § 19, 



<Pg = 2 . »l(aSag — a 2 g) . 



Thus the vector-couple required is 



3M^ 



Referred to co-ordinates moving with the body, <j> becomes p as in § 24, and 

 § 24 gives 



Introducing the value of <p from § 53— i.e., assuming that the earth has two prin- 

 cipal axes of equal moment of inertia, we have 



Be - (A - B)«S« £ = 3M(A - S)Jlqp^ dt . 

 This gives, as in § 54, 



S«e = const. = Q , 

 whence 



£ — — Oa -(- aa , 



