Of CUBIC E^UJTIONS. in 



r _ ? — * 4- • 3- _ V 3 + 1 _ (• 3+ 0* — 2 4-/1- 



^- I _ 2 _|_ v / 3 - v / 3 _I- 2 -2-1-^3* 



v — 3 — a - v' 3 _ • 3 — i _ (• 3 — 0* _ , ,,. 



*Tl — 2 — </$—</ 3 + 1— 2 — 2 V3» 



And the three roots are 2, 2 -f- ^ 3, and 2 — v/ 3. 



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

 two equal roots. 



Example 3. Let the equation be 



x * — yx 2 — 5* + 75 — o« 

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



M = — 16 

 N = + 208 

 772 =4-2560 

 « =4-512 

 Q^rz o. 

 Since, then, Qj= o, it is manifeft that a 9 -b 9 and r will be all 

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

 marked that 1 rr .-• 



n 



Since t = co, we have <£>:= 90 , or <p = 270 : Take <p = 90 : 



Then, 



1 



2 rr tan - zz tan 10 == -7- 

 3 J ^3 



2 = tan (^-f-120 ) =tan 150 = — tan3o° = — 4r 



z = tan f| +240°) =s tan 270 = <^ 

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



a -I — 



v 3 



X ~z 



1 V 3 



1 



« 7- 



v 3 



ff = — 





And" 



