Young — Product of the Squares of the Differences, etc. 745> 

 The square of the middle co-efB.cient here is 



and 4 x the prod, of the extremes, 36;pV+ \QSpqa- 12p^ 



.". the remainder is 81^^ + 12p^, 



which, as might have been anticipated, is the same as the remainder 

 above ; and the remainder would still be the same whatever be a. 

 If we represent the expression Q- - 3PF' by Au^ + Bx + C, and the 

 roots of the equation 



A^ + Bx + C = 



by ^1 and r-,^ then will 



— - 4 — = (ri + r.J - 4ri ro = {r^ - ToJ ; 



and therefore, 



B'-^AC=A\n-r-,)\ (3) 



Hence, the square of the difference of the two roots of the quad- 

 ratic equation Q; - ZPP' = 0, multiplied by A'^, is equal, when its sign 

 is changed, to three times the product of the squares of the differen- 

 ces of the roots of the cubic equation (2). We deduce, moreover, th& 

 conclusions following, namely : — 



1. If ^ = 4AC, that is, if the two roots ri, r-z, are equal roots, 

 then also two roots of the cubic equation (2) must be equal roots, 

 seeing that one, at least, of the differences furnished by the three 

 roots must then be zero. These latter equal roots must be the same 

 as the former (ri, ro) : for, representing one of the equal roots of (2) 

 by r, the expressions Q', and SBF', must each be divisible by (z - ry ;. 

 and consequently, 



Q2 - 3BB', that is, Ax'- + Bx+C 



must also be divisible by {x — ry. But this expression (under the 

 stipulated condition, namely, the condition Ti = r2), is divisible by no 

 quadratic factor other than (^ - ri)- ; therefore, a? - r and x — ri must 

 be identical : hence the equations P = and ^ - 3PF' = must 

 have the same pair of equal roots. [When all the roots are equal, 

 Q^ = SPP' ; and there is no remainder]. 



2. If B'^>4AC, that is, if the roots Ti, r., of the equation 

 Q- - 3PP' = are real and unequal, the sign of B' - 4A C will be 



jjhis ; and therefore the sign of the product of the squares of the 

 differences of the roots of the equation P = 0, or (2), must be minus ; 

 which can be the case only when P = has a pair of imaginary roots. 

 Whenever C is minus (the co-efficient A being 2^lus), the sign of 

 B-- 4AC will necessarily he phis; as also when C is zero. The sign 

 must also be plus whenever A and C are both minus ; since if, under 

 these conditions, B'^ - 4A C could be minus, the roots of Q* - 3PP' = 



