a 



:iU\'dms\A •6WO\j;o\»Ap Asg^-i^^^sj^^ i>$,rt^v>Vi^ aw ^O 0^*: 



cmoi^onSe>b!!aw visa fUM^fKi^ro of^d rln?' JG/ft svoilad ? 

 . On Vontinued Fractions in Quaternions. By Sir William 

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

 ^f^xofessor of Astronomy in the University of Dublin, and Royal 



-)sO'ifb<^" Q2J ,r;T l[Coiitiimed from p. 238.] uf oHirr on 'vj« T irf<5 



7^^^ I ^HE geometi'ical theorems stated in recent articles of this 

 ' » i paper, although perhaps not inelegant, cannot pretend 

 to be important : indeed a hint has been given (in a note) of a 

 quite elementary way, in which they may be geometrically 

 demonstrated. But I think that the analytical process, by which 

 I was led to the formulae of art. 5, whereof the geometrical state- 

 ments of art. 6 are in part an interpretation, may deserve to be 

 considered with attention, on account of the novelty of the 

 method employed : and especially for the examples which it sup- 

 plies of calculation with biquaternions. '^' '^^ ^^''''\' 



8. After obtaining the result already published'in this Maga- 

 zine (compare the number for May, 1852), for a certain conti-. 

 nued fraction in quaternions, namely that if 'q^' . 'jl^* *^" '''\ "*;^ 



^ £ aB ogm ^i gniv/or/ 



i^ == ( _ y:^uiu^^jj} lo ifidj ^§.(1 81 iM# 

 ^ ^fxmfilt 9iii bt if yffjp 1 * \a-{-/ ']'- p^ifisqoiq 3ib 1<) amoS 

 ^™>ltef9'i fBor'OfirjjfT tn'.nifq(fR t ixfd .^rli'rfi'j ariilijiffiS ,^i^'^ 



iifd tl'SOt bnB u^-^y \c—u'/ ' 5sk|?m liMi 'bcft'^ 



^ "trf- '■'''' L c ix. ^ i.- , „.• ^ lUdmaasT 9d^ '!y'itJ• 



M*,, M^' bemg roots or tne quadratic equation , j , 



u^.\.ua=^b', ..... . '. '. . (3) 



and aftef 'Mice deducing the theorem (given in page 303 of the 

 Philosophical Magazine for October, 1852), that for the case of 

 real quaternions, and of unequal tensors, , , . . 



'3- ». =< if T„" < T«'f 1 t:r:'^^ ."^m 



it was obvious, as a particular application, that by changing 

 a, b, a, u', u", u^, to a, p, pQ, p', p", p^, and by supposing these last 

 to be six real vectors, among which /8 is perpendicular to all the 

 rest, I might write ^q^'-^q io '{^olf;Cf6 



^^^udiloj. ... . o —o" p —p" , Off jfs bfiB J qirfg 



089dnonoitj^tii/-^=^''"^^— rrP"^ • vl^rsBe am^ 



V- f.f.s '.;:.;. A P^ — P P0~P ..:.. J. |:...,.j . 



and ultimately, 



p^=/, ifTp"<V, ...... (6) 



* The numbering of the equations commences here anew. 



t This result holds good also in ordinary algebra, and even in arithmetic : 

 but in applying it to quaternions, the order of the factors must be at- 

 tended to. 



