I02 A NEW SOLUTION 



Th e two forms differ from one another only in the figns of 

 their terms. The firft and third terms, as well as the fecond 

 and fourth, have always unlike figns in the firft form ; but al- 

 ways like figns in the fecond form. This property refpedting 

 the figns of the alternate terms, by which the one equation is 

 effentially diftinguiflied from the other, 1 fhall denominate the 

 " CbaraEteriJllc of the form''' 



5, I PROCEED, now, to fhew, in what way any propofed cu- 

 bic equation may be reduced to one or other of the two forms. 



I,ET the propofed equation be, 



.V -I- Ax' -f- Bj>r + C = o, 

 where A, B, C denote any given coefficients, pofitive or nega- 

 tive. I affume x z=. j^-^ : (i and b being indeterminate quan- 

 tities, and z a new unknown quantity. And it is to be obfer- 

 ved, that the fuppofition of x ■=. ^T^ is always poffible, provi- 

 ded a be not equal to h : for if n be not equal to ^, a value may be 

 affigned to z, fuch, that the fraiflion ^T- fhall be equal to any 

 number whatever, pofitive or negative. But '\i a -=1 b, the va- 

 Itte of y^r^ is not altered, whatever number z may denote. 



Having fubftituted, and taken away the denominators, we 



get, 

 (rt-fz)'+A(«+z)'(^ + z) + B(« + z)(^+z) = -|-C(Z'+z)^ = o. 



The 



