Of CUBIC EQUATIONS. 113 



Applying this rule to our example, we have: 

 for the two equal roots, x zz - irr ^-2 — c 



*• , 7 n 312 J 



for thp third root x — ~~ 8lQ2 ° ~ 2I2 " 2 — ~" 2949t2 — ~ 1 

 tor tne tnira root, x _ _„ 4688 + 2I2992 — +98304 — 3 



14. Example 4. Let the equation be 



x % — yx z -f* 1 8a: — 18 zz o. 

 Here A = — 7 ; B = + 18; C == — - 18. Therefore, 



M = -j- 7 



N =-25 

 m zz — 72 

 » = — 26 



Q^= + 7^. 



So that we have here Cafe II. 



— ^ai6 — 3v /6 



-/216 ■ 3) /G 



= + 



.73 



54/6 



54v"S — 73 __ 2 y/g— 1 



Therefor,, ^ = l/*fift = ^ 



Hence z 



vr 



I -f-T _^ 



,3/1 — T 2 »/6 



+ V7+-T 



+ 



l£_j L_ 



-___L1„ , _i_ - -26+3 ~ -23 - J 3- 



2 ^/ 6 ' 2 y/ 



And 3 is the only root of the equation. 



15. When r is a furd as -A7, tne value of z (in Cafe II.) al- 

 ways involves radicals of this form, y f /TV j out of which 



the root may fometimes be extracted ; and fo the value of z will 

 be exprefTed by a furd of the fame kind as r. 



Part I. P The 



