CUBIC EQUATIONS. 577 



Example 1. Let x z — 7x + 6 = 0, the roots are 1, 2, — 3, q = 7, r = — 6, 

 4tf s = — 1372 and 27r 2 = 972 .-. — 4q s — 27r 2 = 1372 — 972 = 400, which is 

 the square of 20. 



Example 2. Let x 3 — 97x + 264 = 0, the roots are 3, 8, — 11, q = — 97, 

 r = — 264. Hence — 4q 3 — 27r 2 = 1768900, which is the square of 1330. 



Example 3. Let x 3 — I9x + 30 — 0, the roots are 2, 3, — 5, q — — 19, 

 r = — 30. Hence — iq 3 — 27r 2 = 3136, which is the square of 56. 



Since this function of the coefficients V— (4q 3 + 27r 2 ) plays an important 

 part in the theory of cubic equations, I propose to distinguish it by the symbol <£. 

 We have therefore </> = product of differences of roots. As a verification, let us 

 resume the three last examples. 



Example 1. <P was found = 20. Roots were 1, 2, — 3, .\ their differences 

 (taken in order), — 1, 5, — 4, the product of which three numbers is 20. 



Example 2. <f> was found = 1330. Roots were 3, 8,-11, .-. their differences 



- 5, 19, — 14. And 5 x 19 x 14 = 1330. 



Example 3. </> was found — 56. Roots were 2, 3, — 5, .-. their differences 



- 1, 8, — 7. And 8 x 7 = 56. 



These theories afford an easy solution of any cubic x 3 + qx — r = o, which 

 has a root of the form a + Vb ; where a and b are either integers or rational 

 fractions, and \fb is a Surd in its lowest terms. 



If a + Vb is one root, the other roots will be a — Vb and — 2a. The equa- 

 tion having these roots will be 



x s _ (3 a 2 + b)x + 2a(a 2 - b) = o 

 whence 



q = - (3a 2 + b) and r = - 2a (a 2 - b) . 



Let us calculate the value of or V — ^4q 3 + 27r 2 ) in this equation. First, we 

 have — q z = (da 2 + bf and r 2 = Aa 2 (a* — 2a 2 b + b 2 ). Therefore we have, 



- q z = 27 a 6 + 27 a*b + 9 a?W + b 3 

 .-. - 4g 8 = 108 a 6 + 108 a*b + 36 a?b 2 + 46 s 



- 27r 2 = - 108 a 6 + 216 a 4 Z> - 108 a 2 b 2 



... _ 4 2 3 - 27r 2 = * + 324 a l b - 72 a 2 b* + 46 s 



46 (81 a* - 18 o?b + b 2 ) 

 .-. >/_(4g 3 + 27r 2 ) = ±2\/j(9a 2 -6). 



But since in this equation we know the roots, a much shorter way of finding 

 </>, is to take the product of the differences. Since the roots are a + \fb, a — */b, 

 — 2a, their differences, taken in order, will be 2\/b, 3a — Vb, —3a — Vb, the 

 product of which gives = 2Vb (b — 9a 2 ) the same as before. Now, by hypo- 

 thesis a and b are integers or rational fractions, therefore 2(6 — 9a 2 ) is rational, 

 and therefore (p = Vb multiplied by a rational quantity. 

 vol. xxiv. part in. 7 R 



