431 



plane, the imaginary units become all equal to each other, 

 and may be denoted by the common symbol i ; and the for- 

 mula (r) agrees then with the known relation, that 



TT— R + TT — r' + tt— r" + . . . +7r— R^"-^) zz 27r. 



Again, let r, r', r" be, respectively, the representative 

 points of any three quaternions Q, q', q", and let b^, r^^, r^^ 

 be the representative points of the three other quaternions, 

 QQ'j q'q", Qq'q", derived by multiplication from the former ^ 

 then the algebraical principle expressed by the formula (q) 

 may be geometrically enunciated by saying that the two 

 points R^ and r^^ are the foci of a spherical conic which 

 touches the four sides of the spherical quadrilateral rr'r^r^^^; 

 and analogous theorems respecting spherical pentagons and 

 other polygons may be deduced, by constructing similarly 

 the formulae (q'), &c. 



In general, a quaternion Q, like an ordinary imaginary 

 quantity, may be put under the form, 



Q = /i(cos0 + (— l)^sin0) =.w + (— l)^r, (s) 



provided that we assign to (— 1)^ or v^ — 1 , the extended 

 meaning (l), which involves two arbitrary angles ; and the 

 same general quaternion q may be considered as a root of a 

 quadratic equation, with real coefficients, namely, 



q^-2m;q + |u' =0, (s') 



which easily conducts to the following expression for a quo- 

 tient, or formula for the division of quaternions, 



Q-l q" = — = 5 q", (s") 



I' Q ft 



q" 

 if we define q~^q" or — to mean that quaternion q' which 



gives the product q", when it is multiplied as a multiplicand 

 by Q as a multiplier. The same general formula (s") of di- 

 vision may easily be deduced from the equation (o), by writing 

 that equation as follows, 



