296 PROFESSOR TAIT ON THE ROTATION OF A 



This gives at once the following results, which are necessary in the elimination of 

 o- by differentiation, 



sr 2 = — 1 , <sr = ni-nr , 



■srir = ni , Or = — 71-zr , 



1 x 



sr = — ?rw . 



Also, because 



S?V = , 

 we have 



(ai + h*y = - (a 2 + b 2 ) . 



Differentiating (41), and simplifying at every step by the above auxiliary equa- 

 tions, we have 



q = (ai -f bsr)q 

 q=- (a 2 + V)q + 1>*2 

 q= — (a 2 + b 2 )q — bn 2 vrq + bn (rtzr — bi)q 



q — — (a 2 + V 2 ) q — (bn 2 — fata) irq + (bn 2 — bna) ( —k + b J q — b*n( — a + - a- J q 



— - (a? + b 2 )q - (hi 2 - 2bna + ba 2 + b*)-kq + b 2 n 2 q . 



Eliminating irq from the last equation by means of the second, we have for the 

 determination of q the linear equation of the fourth order with constant co- 

 efficients 



q + [2 (a 2 + I 2 ) + n 2 - 2na]q + [(a 2 + b 2 ) 2 + (a 2 + b 2 ) (n 2 - 2na) - b 2 n 2 ]q - (42). 



Assume, as a particular integral, 



q = Qe mt , 



where Q is an arbitrary, but constant, quaternion, and e is the base of Napier's 

 Logarithms. Then we find for m the equation 



m 4 + [2(« 2 + b 2 ) + n 2 - 2na]m 2 + (a 2 + b 2 - na) 2 = , 

 or 



m 2 + a 2 + b 2 — na = zk x/ — mV . 



Hence m is imaginary, so we may write 



m = [IaJ — 1 , 



and our equation gives 



[i 2 ± i*n = a 2 + b 2 — na , 



whence 



By § 50 this may be written 



