no A NEW SOLUTION 



Since M + N r= 3 ( i + A + B + C) : it is obvious, that when 

 M -f- N = o, we fhall alfo have i +A+B + C = o; and ib 

 one root of the equation is unity. It is manifefl too, that, in 

 this cafe, the transformed equation for z becomes fimply, 



32' — I :r: o : whence z ^ + -r-; and confequently the other 

 two roots of the equation are, x =z ^, and x =z ^^. 



Or, writing for a and /', their values, , _ (. and , q , the 



two roots are, x = — f-^ ^and x zz y — j^^ 



12. Example 2. Let there be propofed 



x ' — 6x' -i- 9X — 2 zr o. 



Here A = — 6; B = + 9; C = — 2: And hence, 



M = o 



N = + 6 



«? = + 54 



« = + 18 



Q= - 108. 



And we have here again Cafe I. 



»- + 3 

 b = + I 



T — I 



Therefore <p — Arc. tan — i zz 145°. therefore, 

 2 = tan - n tan 45 ° zz i , 



z = t.(? + i2o°) = t. 165° =-1.15° =-(2-^3), 



2: = t. g-}-24o'j = t.285° = -t.75°=-(2+/3); 



Therefore, x =: -M^ = - = 2 ; 



