[ 303 ] 



XLVI. On Continued Fractions in Quaternions. S?/ Sir William 

 Rowan Hamilton, LL.D., M.R.I.A., F.R.A.S.^-c, Andrews' 

 Professor of Astronomy in the University of Dublin, and Royal 

 Astronomer of Ireland"^, 



[Continued from vol. iii. p. 373.] 



3. TT results from what has been shown in the two former 

 J- articles of this paper, that, whether in quaternions f or 

 in ordinary algebra, the value of the continued fraction. 



/ b V 

 " \a + J ' 



■ (1) 



may be found from the equation 



^=^. P) 



u^-v! 

 where 



«.=«"'?£^«-'' (3) 



or from the expression 



u={\-vy{u!'-vic'), (4) 



if m' and m" be two unequal roots of the quadi'atic, 



u^-\-ua = b (5) 



If, then, a b c u' u" be five real quaternions, of which the three 

 last are unequal among themselves, and the two latter have 

 uneqvial tensors, 



Tm'>Tw", (6) 



we shall have the following limiting values : 



Tv^=0, v^=0, u^=u". ... (7) 



We may then enunciate this Theorem : — If the real quaternion c 

 be not a root of the quadratic equation (5) in u, the value of the 

 continued fraction (1) will converge indefinitely towards that one 

 of the real quatei-nion roots of that quadratic, which has the lesser 

 tensor. If the quaternion c or u^ be a root of that equation, it 

 is clear that the fraction will be constant. 



September 21, 1852. 



[To be continued.] 



* Communicated by the Author. 



t The writer has again to regret that unforeseen causes of delay have 

 occurred to retard the jjublicatioii of his Volume of Lectures on Quater- 

 nions, of which, however, lie hopes that the value will be found to have 

 been increased, by the additions which he has inserted. 



