TKACT 2S.

AND INFINITE SERIES.

93

when the parts are equal to each other, or each equal ±ar }and therefore the greatest product is equal to (j.r) z ortherefore, if at 1 be equal to or greater than — j-p, the con-dition which supposes that zy is = — j-p, is possible, andthe formula ought to express the root by real quantitiesonly; otherwise not: but ±x z , or ~r z , which is the samething, is always less than —4 p in the first 13 cases of thetable of forms ; and therefore, in all these cases, which arethose in which ($p) 3 is greater than (~q) z , or all those whichhave three real roots, the formula ought to exhibit the rootwith imaginary quantities, as we have before found to hap-pen ; the 4th and 13th cases only excepted, in which (ip) 3is =: and therefore the quantity \/{b z — a 3 ) vanishes,

and two of the roots are equal.

48. Thus then the real cause of this circumstance is mademanifest, and it is found to be the necessary consequence ofthe arbitrary hypothesis that was made, which is found to bepossible only in certain cases. So that we cannot expectthe formula to exhibit a real quantity in the other cases,since an impossible hypothesis must needs lead to an absurdconclusion.

49. The other two roots ± — in their ge-

neral state, appear in an imaginary form ; but on the sub-stitution of numbers for the letters in any example, theycome out real, or imaginary quantities, in those cases inwhich they ought to be such. For s being = g -f

and d = g — +h, according as the roots are all real or

only one is such; and — = — g = — always half the

one real root, we have zt */+ h according to the said

two cases; and consequently 3 == y' ± Zh, a real ot

an imaginary quantity, according as the roots are to be realor imaginary.

50. The first root r being found from the formula

l/\b + + « 3 ),] +M* — + « 3 )], or by any other