431 



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

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

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



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



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

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

 be the representative points of the three other quaternions, 

 QQ', 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 formulas (q'), &c. 



In general, a quaternion q, like an ordinary imaginary 

 quantity, may be put under the form, 



Q = fi{cosd + (- 1)4 sin6) = «/; + (— l)*r, (s) 



provided that we assign to (—1)*, or y^ — l, 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'-2«/;q + ^^ = 0, (s') 



which easily conducts to the following expression for a quo- 

 tient, or formula for the division of quaternions, 



' Q )U 



q" 

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

 Q 



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

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

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

 that equation as follows, 



