is 
ee Dr. Wantne on the 
minator in the firft example, and the denominator in the 








cond, and every diftinét irrational- quantity contained in the 
relpeclively =o; and alfo every. diftin& irrational quantity. con 
tained in the numerators =o. Suppofe « the lealt root affir~ 
mative or negative (but not =o) of the abovementioned re~ 
fulting equations ; then a feries afcending according to: the aie 
menfions of x will always converge, if the value of eh 
is contained between a and —a; but # # be greater 
than # or —a, the abovementioned feries will diverge.. Let 
g be the greateft root of the abovementioned refulting equa- 
tions; then a feries defcending according’ to the reciprocal di-7 
menfions of x willjconverge, if.w be greater.than = 7; but, if 
lefs, not, . When impoffible roots 4 =& b/ SI are contained in 
the equations, an afcending feries: will converge, if x be lef. 
than the leaft root = #, and = (¢—4), and = (¢+6); of 
more generally, if w be lefs than. the leaft root == a, and vel 
at an infinite diftance 2, be infinitely lefs than | 
/(n—t Gi 2-2 An 3 
2qan—-2.N» ar—2 b?+ DS 6 ‘wy a . > < 2 gt—4 b+. &c.. aTé 
3 ‘ 
~ 7 . ra > eo. 



(a*+5°)" 
a defcending feries will always converge, when w is greater) 
than the greateft root of the refulting equations 3, and a*—*,,_ 
when # 1s fidlite, is infinitely Sapa than (4+ 6)" and (4 = 4)"57 
n-~ n—2 


or more generally than 2a"-22”. 
b 
& 
oe a’—4h* — &e ° 
