432 



, , , , . , , X 1 w — ix—jy—hz , ,, 



(e. + ^^• ^jy¥ k.y = ^.^,./;.^,. ; (oO 



or it may be obtained from the four general equations of 

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

 multiplicand, namely, u/, x', y' , z' , as the four sought quan- 

 tities, while w, X, y, z, and w", x" y", &", 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 i^ having 

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

 proposed real power; and multiplying the amplitude 0, in- 

 creased or diminished by any whole number of circum- 

 ferences, by the exponent q : thus, 



ill (cos Q -f- i^ sin 0))« — fit (cos q{9 + 2mr) + /„ sin q{d + 2mr)), (t) 



if q be real, and if n 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, 



(ju(cos d + ^Ksin 9))* = fi*[cos (- + rnr) + i^sm (^ + wtt) j, (t') 



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 9 ■=nr, this formula (t') becomes 



(-|u)» = ± ju*«B, or simply (-fif = fxH^, (t-'O 



the direction of i^ remaining here entirely undetermined ; a 

 result agreeing with the expression (l) or (l') for V^ — I. In 

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

 are included in the expression, 



1' = cos -g- + ^K sm — g-, (x"') 



