420 The InverJe Method 
found; therefore thefe values muft be equal to each other; 
and converfely, if thefe values be equal, the given equa- 
tion mult have two equal roots; and if the conditions 
above mentioned take place, viz. that’ , and m, be both 
even numbers, —- A negative, and — 8 pofitive, then 
there will be two pairs of equal roots, Make them equal 
to each other, then by reduétion it will eafily appear 
u—m 
m 
that m—n | A” Ky n—m 
eet ns, sath Dg TE = 0; hence, 
nN i) 
by making this expreffion pofitive, or negative, according to 
circumftances, the number of poffible roots may be obtained. 
1, If be an even number, and — B negative, it is 
evident that the equation has two real roots, If ” be 
even, and m odd, and —B affirmative, and at the fame 
n—m ‘ 
m—n | a” 
time Ba. ed 
> Brn be negative, 
n—Im mit Z 
n 
then there will be two pofitive roots; otherwife uone. 
For the firft part of the above expreffion is in this cafe 
negative, and the other part pofitive; but the firft part 
muft be greater than the fecond, if the roots be real; 
becaufe if A vanifh, all the roots are impoffible; hence 
the conclufion is clear, 
Let n, and m, be even numbers, and — 4, and —B, 
pofitive quantities; it is evident that the equation can 
have no real roots; for in this cafe no quantity fubftituted 
for x can make the refult = 0. But if — B be affirmative, 
u—™M 
t : m—n 
and — A negative, and at the fame time 
——— x B"~” be affirmative, then there will be four 
™m 
real roots, otherwife none, For, from what has been faid 
above, there will be two pairs of equal roots when the 
above 
