A New Method of Resolving certain Equations. ii7 



Put a' =»•+«, ) Then, 



« c'=r=»-i-3ar2-f 36r+c3 so a'd^h'". 



mz^-\-Sa'mz^-\-3h'mZ'\-c'm~o 

 z ^ 4- 3a^OTz3 hP ih'mz + c^m = 



[i — m.z'. 

 Assume (a'm)^ =5'm 

 Then also (a'w)3=c'm,- For, (a'wi)* (i'm)' 6'^ 



Substitute values; 



[=Vl -m.g 



6' 



(«'+V(6'— «'^)a'.^=-&', ;?=- === 



a'+W{b'-a'^)a' 

 r^-\-2ar-\-h 



From this we derive the 



Rule. 



To find a?, when x^-\-Zax^-\-Qbx-\-c=o ; 



(A) Find r in the quadratic, 6-a=.r2+c-a6.r+ac-Sa=«, 



(B) Put z =- 



And X —z-\-r. 



In those questions which are enfibraced by the irreduci- 

 ble case of Cardan, it will be seen that the value of r is 

 imaginary. For th en, if ■r^4- 36x+c=o, hr'^-\-cr — h-=o, 

 whence 26r4-c=v'c^ + 46 ^^ ; which gives an imaginary 



value, whenever 6^ is negative and exceeds — . When 



these two quantities become equal, the imaginary part of 

 the ex pressio n vanishes ; and, in the general solution, we 

 have 2b — a'^.r-]rC — ah~o. (C) 



