of Central Forces. | 422 
above expreffion —o0, wherefore by making it affirmative 
there will be four real roots; That the firft part of the ex- 
preffion muft be greater than the fecond will be evident 
from obferving, that if A= 0, then all the roots will be 
impoffible, — 
. MALS % 2—m 
2. If both 2 and m be odd numbers, ‘and 
n” 
An —— x 3B" affirmative, then there will be three 
m ; r 
poflible roots; otherwife only one. Becaufe n is an odd 
number there muft be one real root: and if the above 
expreflion = 0, there will be other two roots equal, But 
if A.and B be taken fuch as to make it affirmative, there 
will be other two real roots. For nm is an even number, 
therefore the firft part of the expreflion is pofitive; and 
if A =o, there will be only one real root, and the above 
expreffion would in that cafe be negative. Therefore, &c, 
If 2 be odd, and.m even, and A, and B, have the fame 
fign, then the equation will have only one real root; this 
requires no proof, But if they have different figns, and at 
n—m 
n—n n 
es IN 
the fame time —— —- x Ar— 
ne 
x B"-” have 
a contrary fign to B, then the equation will have three 
real roots; otherwife but one.—If the equation have this 
ui—m 
form, viz. x” + 4«%” — B= o, then ssa ae 
? : i ee nim 
“ 
will be pofitive, and it muft be greater than ——— x B"—", 
for reafons given above; therefore the whole is pofitive, 
But if it be of this form, viz. x” — Ax” 4 B—o, ‘then 
the firlt part of the general te) is Biaeitire and the 
fecond part pofitive, wherefore the refult muft be negative, 
