220 
ferences) that the numerator and denominator of the resultant 
fraction satisfy two equations in differences, which are of one 
common form, namely, 
Nee = A eee = Ne 3 Vrwas 
Diaz = » be a a ae 
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 d to be two constant quaternions, these 
equations in differences are satisfied by supposing 
N, = Cq,* + C'q*; 
D; Eq," + E'q:", 
C+C’=0, Cn + Ca =, 
E+H=1, En + E'q=4;3 
Il 
Il 
C, C’, E, E’ being four constant quaternions, determined by 
the four last conditions, after finding two other and unequal 
quaternions, g, and g,, which are among the roots of the qua- 
dratic equation, 
g = qga + b. 
3. By pursuing this track it is found, with little or no diffi- 
culty, that 
where 
gi» Jz being still supposed to be two unequal roots of the 
lately written quadratic equation in quaternions, 
g=qat+ b. 
4, Let the continued fraction in quaternions be 
