220 



ferences) that the numerator and denominator of the resultant 

 fraction satisfy two equations in differences, which are of one 

 common form, namely, 



N a+1 = N x a x + 1 + N-e. i&a. + l , 

 ■IJx+l — Dx a x+1 + JJx-\O x + i. 



And by the nature of the reasoning employed, it will be found 

 that these equations in differences, thus written, hold good for 

 quaternions, as well as for ordinary fractions. 



2. Supposing a and b to be two constant quaternions, these 

 equations in differences are satisfied by supposing 



N x = Cqf + C'qf, 



D x = Eqf + Eqf, 



C + C = 0, Cq x + C'q 2 = b, 



E + E' = 1, Eq-i. + Eq 2 = a ; 



C, C", .E, E being four constant quaternions, determined by 

 the four last conditions, after finding two other and unequal 

 quaternions, q x and q 2 , which are among the roots of the qua- 

 dratic equation, 



q 2 = qa + b. 



3. By pursuing this track it is found, with little or no diffi- 

 culty, that 



where 



„ _ (±Xq. &-& - g -i V». 



ji, g-25 being still supposed to be two unequal roots of the 

 lately written quadratic equation in quaternions, 



q 2 = qa + 3. 



4. Let the continued fraction in quaternions be 



