and Infinite Series. 409 
tities only, otherwife not; ov\r % , which is the 
fame thing, is always lefs than - \p in the firft thirteen 
cafes of the table of forms ; and therefore in all thefe cafes, 
which are thofe in which |pl is greater than |yT, or all 
thofe which have three real roots, the formula ought to 
exhibit the root with imaginary quantities, as we have 
before found to happen; the 4th and 13th cafes only 
excepted, in which |p 3 is = , and therefore the quan- 
tity V b z — a 1 vanifhes, and two of the roots are equal. 
48. Thus then the real caufe of this circumftance is 
made manifeft, and it is found to be the neceffary confe- 
quence of the arbitrary hypothefis which was made, 
which is found to be poffible only in certain cafes. So 
that we cannot expert the formula to exhibit a real quan- 
tity in the other cafes, 11 nee an impoffible hypothefis 
mull needs lead to an abfurd conclufion. 
49. The other two roots - — - ± — V —3 in their 
general Hate appear in an imaginary form ; but on the 
fubftitution of numbers for the letters in any example, 
they come out real or imaginary quantities in thofe cafes 
in which they ought to be fuch. For s being = g + ■ s/ d -b y 
and d — g—V^h according as the roots are all real or 
only one is fuch; and — - — - g- — \r always half 
'1 
the one real root, we fhall have s -f- = s/ according to 
7 the 
