no A NEW SOLUTION 



Since M-fN:=3(i-f-A-f-B-{-C):itis obvious, that when 

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

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

 this cafe, the transformed equation for z becomes fimply, 



32* — 1 zo: whence z zz + ~rz ; and confequently the other 



1 r l 



a * T7~ a — 7/~ 

 two roots of the equation are, x — : — , and x =z —• 



Or, writing for a and 3, their values, , ^_ ^ and , _ q , the 



two roots are, x = ^ jj* _^ and * — w _ y >__ ( ^ 



12. Example 2. Let there be propofed 



x J — 6x* + 9* — 2 = 0. 

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



M = o 



N = + 6 



w = + 54 



n = -f 18 



Q= — 108. 

 And we have here again Cafe I. 



a = -f- 3 



£ = + 1 



r r_: — I 

 Therefore <p = Arc. tan — 1 = 145°, therefore, 



z = tan - = tan 45 ° =: 1 , 



z = t. (| + iao°) = t. 165 = - 1. 15 =— (2-/3), 



2 = t.(|+2 4 o ) = t.285 = -t.75 =:-(2+/3); 



Therefore, x = -7^7 = - = 2 : 



A? 



