AND IMAGINARY ROOTS OR EQUATIONS. 
651 
(81) Thus 
5 10 / 4 \ 
-Z=+2"L-P+fjD) 
=t(a+» 
and accordingly we have proved that — Z is of the form (A + f JD); and consequently, 
since f lies within the allowed limits 1 and — 2, — Z may be used to replace A in the 
system of criteria. 
(82) On examining the composition of Z, it will be found to have a remarkable relation 
to the lower criterion J. 
J we know is, to a numerical factor pres, of the form 
2{(d— e)%{a, b, c)}, 
% denoting, as usual, the squared product of the differences of the quantities which it 
affects ; and Z, it will readily be seen, is of the form 
(£(«, b, c, d, e )) ^^ ajb)C ^ d _ e ^ 
and the squared factor is always positive whatever the roots maybe, for £ is always real. 
Hence the essential part of our rule thus transformed comes to this, that if 
2(£(«, b , c)x(d— ey'j and 2j(£(<z, b, c)) \d- e)~ 4 j 
are both of them positive, then when the discriminant is positive, so that the case of two 
of the five quantities a, b , c, d, e being conjugate and the other three real is excluded, 
and the choice lies between supposing all or only one of them real, we are able to affirm 
that they will all be real. Nothing could be easier than to multiply tests expressed by 
simple symmetric functions of differences of the roots, any infringement of which would 
contradict the hypothesis of all the five letters denoting real quantities ; the difficulty 
consists in discovering a system of the least number that will suffice of decisive tests, 
such that not only their infringement shall contradict the hypothesis of imaginary roots, 
but whose fulfilment shall ensure the roots being all real. This is what has been proved 
to be effected by means of the invariants J, D, A+f JD. 
In the case before us it is clear that when the roots are all real, each of the sums 
above written must be positive and greater than zero. That their being both positive and 
greater than zero is inconsistent with four of the letters a , b, c, d, e being imaginary 
would probably not admit of an easy direct demonstration. 
Z we have seen is only a particular value of the general invariant A+^JD, which 
may be called M, where p is an arbitrary constant limited to lie between 1 and —2. 
(83) It may be well to notice the effect of using as a criterion , in conjunction with 
J and D, the value of M corresponding to either extreme value of In such case, 
supposing M to become zero, it might for a moment appear doubtful to which region 
mdccclxiv. 4 s 
