374 



Now, this expression is positive for every real value of %, provided 

 it be either a complete square, or that it satisfy the condition 



4(A* - SA^Af - 3A 0 A,) > {A,A 2 - 9A 0 A 3 f ... [2] 



and it cannot be positive, for every real value of x, unless one or other of 

 these conditions hold. Hence, when all the roots of P = 0 are real, 

 [1] must be a complete square, or else the condition [2] must have 

 place, and conversely ; so that the condition which must be satisfied 

 when a pair of roots is imaginary, and which cannot be satisfied unless 

 there be an imaginary pair, is 



4(^ 2 2 - 3A l A 3 ){A l * - 3A 0 A 2 ) < (A X A, - 9A 0 A 3 )> ... [3] 



The criterion of imaginary roots for an incomplete cubic, is, of 

 course, but a particular case of this more general condition ; the case, 

 namely, in which A 3 = I, and A 2 = 0 ; for making these substitutions, 

 [3] becomes 



- 124,3 <{9Aoy> or (v 



as at p. 351; A l} A 0> here, being p, q, there. 



It follows from the above conclusions that when the roots of 

 Q 2 - 3PP' = 0 are imaginary, those of P = 0 must all be real ; and that 

 when the roots of Q 2 - 3PP' = 0 are real, two roots of P = 0 must be ima- 

 ginary, and vice versd, unless the roots of Q? - 3PP f - 0 are equal roots ; 

 and equal roots they will always be whenever P = 0 has two equal 

 roots, the equal pair in the former of these equations being the same 

 as the equal pair in the latter. For one of the two equal roots of P 0, 

 namely, x = r, must enter the equation Q = 0 ; so that Q 2 and 3PP' 

 are each divisible by {x - r) 2 ; therefore r, r, are the two roots of the 

 quadratic Q 2 - 3PP'=0. In the case supposed, therefore, [1] is always 

 a complete square :* hence, if the equation P= 0 have equal roots r, r, 



A 2 2 - 3A 1 A 3 , and A? - 3A Q Ar, 

 must be squares ; and 



] A* - 3A 0 A, 



and whether the sign of r is to be positive or negative will be at once as- 

 certained from the signs in the proposed equation, by the rule of Des- 

 cartes, as all the roots must be real. 



* Of the form m?(x - r) 2 . 



