308 THIRD REPORT 1833. 



which gives, when solved as a quadratic equation, 



jc^ = r + s/ {f- — (f), 

 and consequently, 



« = (r + -v^ (f - q^)Y, 

 and therefore 



^ q_^ 1 



^ u {r + s/{r^-(f)Y 



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

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

 arithmetical value of u, we shall obtain the following three 

 values of ^l + v, which are 



a + ■^, a a. ■\- -i-, « a^ + -^ 

 a a a. aa.'- 



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 



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

 = ; that the sum of their products two and two = — 3 g'; 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 f. 



* 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 ia usual to express the roots of the equation .r' — 3 5 a; + 2 r =: 0, by the 

 formula 



x={r^- V(r2-53)}T + {,._ ^(,.2_23-)}f, (i.) 



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 



a;3_3«gfa! + 2r = 0, 



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



(^ _ 2 rf — 27 q^ 3? = 0, 

 which is of 9 dimensions. It is the formula 



*•=«+—, where m = {r + V ix^ — g^) Y' 



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

 the equation a^ — 3 y a: + 2 r = 0. 



