• Of CUBIC E^UJT'IONS. iii 



,- — 3 — ^-v^3 — •'3 — t _ C^/3 — ly _ 



And the three roots are 2, 2 + ^ 3, and 2 — ^/ ^. 



13. I TAKE the following example to illuftrate the cafe of 

 two equal roots. 



Example 3. Let the' equation be 



X * — yx"^ — jj; + 75 = o« 

 Here A = — 7: B = — 5: Crr + 75. Therefore, 



M = — 16 

 N = + 208 

 m zz + 2560 

 « = + 512 



Since, then, Qj= o, it is manifeft that a,l), and r will be all 

 infinitely great. But though a and b be infinite, it is to be re- 

 marked that 7 = -• 



6 n 



Since r =. 00, we have <pzz 90°, or 9 = 270°: Take (p = 90° : 

 Then, 



3 = tan| = tan3o=^ 



z = tan (^ + i20°j =:tan 150° = — tan3o°= — 4- 



z =. tan (^ +240°) = tan 270° = o^ 

 We have then, in the firft place, for two roots, 



^ V3 



I 



x=- p- 



Vi 



And: 



