and Infinite Series. 419, 
76. And by taking other values of b and c } or other 
relations between them, any number of infinite feries may 
be affigned, whofe furns will be given by the two equa- 
tions 4 /b ±c- 4 /b x : 1 ± — — ~ k - ± - 2 ' 5c — &c. And 
2,b 3.bt> 3.6.9A 3 
if Abe very great in refpedt of c, the two firft terms of 
the feries will give the cube root true to many places of 
figures. 
77. Hitherto is concerning one of the limits or ex- 
treme cafes only, namely, when c z — A, or when the 
equation is — q — 2 b. And it has been obferved, that 
the firft general feries for the three roots converges in 
all the cafes of the equation x l — px — q, or x l — 3 ax — lb, 
- in which a 3 is not greater than 2 b z . But a 3 may be any 
real quantity not greater than 2 A, and fo it may be 
either lefs than, equal to, or greater than If. 
78. When, in this equation, a 1 is lefs than A, then c z 
is pofitive, and lefs than A, and the firft feries gives the 
only real root without any change in the figns of the 
terms. And to this belongs all cafes of the equation that 
can fall in between the 1 3th and 1 6th formulae in the 
general table in Art. 30. 
79. If a? be = A, then c — o, and the three firft feries 
give %-I/b = 4/4 q for the greateft root, and *- 4 / b for 
each of the lefs roots. The fame as at the 1 3th form in 
the general table Art. 30. 
Vol. LXX. 
K k k 
80. When 
