308 THIRD REPORT — 183^. 



which gives, when solved as a quadratic equation, 



ti' 



= r + >^ if — (f), 



and consequently, 



and therefore 



^^q_^ 9' 



u {r + v/ {r^ — (f')Y 



If we call 1, «, o?, the three cube roots of 1, or the roots of 

 the equation s^ — 1 = 0, and if we assvime a to represent the 

 arithmetical value of u, we shall obtain the following three 

 values of ?< + v, which are 



a + -^s a a ■] 2_ « «^ -| 2-. 



a a u aw' 



These values, though derived from the solution of an equation 

 of six dimensions *, are only three in number, and form, there- 

 fore, the roots of a cubic equation. A little further inquiry will 

 show that they are the roots of the cubic equation 



x^ — 3qx + 2r=0: 



for it may readily be shown, in the first place, that their sum 

 = ; that the sum of their products two and two =: — 3 q; and 

 that their continued product = 2 r; or in other words, that 

 they are the roots of an equation which is in every respect iden- 

 tical with the equation in question-}-. 



* There are six values of u, in as much as the values of u and v are inter- 

 changeable, from the form in which the problem was proposed ; but there are 

 only three values of u + v. 



t Since q ^ i 



it is usual to express the roots of the equation x^ — 3q x -\- 2r = 0, by the 

 formula 



x={r+ V(r^-q^)}^ + {'•- V(»-=-?^)}i (1.) 



which is in a certain sense incorrect, in as much as it admits of nine values 

 instead of three. The six additional values are the roots of the two equations 



x^ — 3ctqx + 2r = 0, 

 x^ — 3a'^qx + 2r=0, 

 and the formula (1.) expresses the complete solution of the equation 



(«« — 2 rf — 27 </3 a;3 = 0, 

 which is of 9 dimensions. It is the formula 



« = M + -|-, where u= {r + V (r^ — q^) }i 



and has the same value in both terms of the expression, which corresponds to 

 the equation x^ — 3qx + 2r — 0. 



