580 MR TALBOT ON SOME MATHEMATICAL RESEARCHES. 



.-. a = 7, and the roots are 7 ± \/13, — 14. The same rules apply when the 

 coefficients are fractional, of which I will give some examples. 



Example 8. Let x 3 — -~x - — = 



TT -id 1 * ■*- TX A 1 OOiD 



Hence # = — -r- and r = -j- . Hence — 4g i ' = 



4 _ 4 ' * 



67 . . 108 27 , . 3 

 ^= T and = 2 



15 3 



^ - - 4 = ~ dl * • • «"- 2 ' 



_ 2 3267 . 2 108 27 , . 3 /5 

 and ^- .-. </> 2 ^- = - and = 2 • V3 



Hence 6 + ^ = 3— -j- = — - = — 3a 2 .-. a == 



and the roots are „- ± \/3, — 1. 



Example 9. Let x 3 - ^ a; + y^. = . 



Here — 4<? 3 — 27r 2 comes out a fraction, of which the numerator is 209952 and 

 the denominator 3 . 15 3 . First let us take the latter, which may be written 

 3 4 . 5 3 , but omitting the square factors, this reduces itself to 5. The numerator 

 is 5 times divisible by 2, the last quotient being 6561, which = 3\ Hence 

 omitting square factors the numerator = 2. Therefore 



(p 2 or b = = . 



. . 2 26 4 Q . 2 



Again b + q = 5 - T& = - g = - 3a 2 .-. a = g . 



Hence the roots are * — J% , — «• 



The same rules apply when the roots are imaginary of the form a ±V — b, pro- 

 vided that -\/£> is a surd m «7s fow^ terms. 



Example 10. Let # 3 — 22# + 84 = 0. Here ^ = — 22 r, = — 84, whence 

 — 4<7 3 — 27r 2 is a negative quantity — 147920. This number is 4 times divisible 

 by 2, and then once by 5, the last quotient is 1849, which is the square of 43. 

 Hence $ 2 — 2 4 . 5 . 43 2 taken negatively. And omitting square factors, <£ 2 becomes 

 b = — 5. Hence b + q ■ = — 5 — 22 = — 27 = — 3a 2 .-. a = 3, and the roots 

 are 3 ± \/ — 5, — 6. 



Example 11. Let £ 3 — 68 # + 320 = 0. Here — 4# 3 — 27r 2 is negative, and 

 = 1507072. This number is divisible 8 times in succession by 2, then once by 7, 

 and the last quotient is 841, which is the square of 29, therefore <f> 2 = 2 8 . 7 . 29 2 

 taken negatively, and omitting square factors <£ 2 becomes b = — 7. Hence 

 b + q = — 7 — 68 = — 75 = — 3a 2 .*. a = 5, and therefore the roots are 

 5 =h a/^T" , — 10. 



