372 Sir W. R. Hamilton on Continued Fractions in Quaternion 



Let q^ and q q be any two assumed quaternions ; then 



Ug+i + q 1 = b(a + tt i )- 1 + q 1 = (b + q x a + g,«;) (a + u)-\ 

 ^+i + qo=b{a + u :t )- 1 + q % ={b + q 2 a + q 2 u x )(a + uJ- 1 > 

 u*+\ + qo _ b + q^a + q^ _ q~ l b + a + u, v _, 

 u*+i + q l b + q^ + q^ ?2 q-^b + a + u r CJl 

 If therefore we suppose that q v q 2 are roots of the quadratic 



equation 

 which gives 

 we shall have 



and finally, 



q~ — qa + b, 

 q~ } b + a = q, 



"«+! + & M * + gg -, 



u.v+i + q x % + Q\ ' ' 

 M^ + g 2 _ , r Up + qg _, 



M r + (7l _92 W + ?1 ?1 



2. It was in a less simple way that I was led to the last 

 written result. I assumed 



( h Y 



11, ,= I ) c, 



and treated this continued fraction as a particular case of the 

 following, 



By changing c to ^iJ— , I obtained the equations, 



N', + 1 =N', ^ + N", b t , N", +1 = N' W 

 D' r+ , = D', a v + V", b r , D",, + , = D ? „ 



with the initial conditions 



N' 1 = 0, N" 1 = l, D', = l, D" 1= =0, 



which allowed me to assume 



N' =l, I)' =0. 

 Making next 



a =:a, b x =b, 

 there resulted 



N,= N'> + c) + N',_ A D,=D',(« + c) + V'^b, 

 W, + , = N' a a + N',_ A D', + j = D', « + D',._ i'. 

 This led me to assume 



N ',. = ¥i + mg*2, D' r = /Vy '', + m'q% 

 Qt = a + q~ 1 b, q^-a-\-q- l b, 



l + m=l, lq l + mq< 2 = 0, l' + m' = 0, l'q l + m / q i =l ; 



