442 Dr. hutton o;z Cable Equations 
126. Of which the feries in the firft and third de- 
note the only real root of the equation when c 1 is pofi- 
tive, according as c is greater or lefs than b, which root" 
call x ; and the feries in the fecond and fourth forms 
denote the greateft and leaft roots of the equation when 
c 2 is negative, which roots call r and r refpedtively. 
Then by adding and fubtradting the firft and fecond, as 
alfo the third and fourth, there refult thefe four equa- 
tions; 
5 , v- a !/K. t _ r _ 2.S.8.II.14.17. 
4v ' * 3.6.9. i2^ 4 3 . 6 . 9. 12 . 13 . 18. 21 . 24# 
20c 
Sec. 
r - x = 4 byb x : ~~ri + - ~ ; " ■ 8 . — j . 1 - 4 -! + Sec. 
" v 3.6.9.12*15.180° 
x - r = x : _L + V' 5 ; 8 ,Ilj 4 V + See. 
x + r 
- ii 
3 / .1 
3.6.9.12.15 
2 . 
2.5.8.11.14. 17 b 6 
+ 8cc. 
3.6.9 C z 3.6.9. 12 . 15 . 18. 21 C e 
127. And hence, by equal addition or fubtradtion, we 
find thefe two different expreffions both for the greateft 
and leaft roots of a cubic equation in which c 2 or b 2 + a 3 
is negative, namely, 
r = — x + 4 J/b x : 1 - 2 - 3 - 8 - ir - I 4 - 1 7 - 20 ^ 2 ee. or 
3.6.9. I 2* + 3 .6.9. 12. 15. l8. 21. 245 
R = X + 
3 /7 2 C 
3 / 7 . x . 
+ + &c. 
3.6 b 3.6.9.12.15.18^ 
r=x-^ x ; A + ^ 5 . 8 -^ + or 
3 3.6.9.12.15c 4 3.6.9.12.15.18.21.24.27c 
r = - x + 
4 * 
3.6.9c 3 . 6 . 9 . 12 . 15 . 18 . 2 IC° 
where r is the greateft, and r the leaft root of the equa- 
ti on 
