432 



/ . • ■ • . I \ 1 w — ix~jy—hz , ,. 



(e. + .X- +^y + kzy- = ^.^^.^y^^.3 ; (o') 



or it may be obtained from the four general equations of 

 multiplication (d), by treating the four constituents of the 

 multiplicand, namely, ti/, x', y' , %', as the four sought quan- 

 tities, while w, X, y, z, and w", x" y", z", are given ; or from a 

 construction of spherical trigonometry, on principles already 

 laid down. 



The general expression (s) for a quaternion may be raised 

 to any power with a real exponent q, in the same manner as 

 an ordinary imaginary expression, by treating the square 

 root of — 1 which it involves as an imaginary unit ij, having 

 (in general) a fixed direction ; raising the modulus /x to the 

 proposed real power; and multiplying the amplitude 9, in- 

 creased or diminished by any whole number of circum- 

 ferences, by the exponent q : thus, 



(fi (cos d + i^ sin 6)y = ju^ (cos q{e + 2tnr) + /„ sin q(e + 2nir)), (x) 



i£ q be real, and if « be any whole number. For example, a 

 quaternion has in general two, and only two, different square 

 roots, and they differ only in their signs, being both included 

 in the formula, 



(jii (cos 6 + insin 0))* = fv{cos (- -l- mr) + ?b sin (- + mr)j , (x') 



in which it is useless to assign to n any other values than 

 and 1 ; although, in the particular case where the original 

 quaternion reduces itself to a real and negative quantity, so 

 that -znr, this formula (x') becomes 



(— ^)>=: ±ju*'r. or simply (-m)' = fJ-^k, (x-'O 



the direction of 4 remaining here entirely undetermined ; a 

 result agreeing with the expression (l) or (L')for \/— I. In 

 like manner the quaternions, which are cube roots of unity, 

 are included in the expression, 



1 » = cos — h ?R sui — ^, (x"0 



