and Infinite Series. 
443 
tion * 3 - 3 ax - ib or x l - c z + b z . x — a b, and x 
0 
the only real root of the equation x l + 3 'c 1 — b* . x = ib 
in which, as well as in the above feries, c 1 denotes a 
pofitive quantity. 
128. And hence it can no longer be faid that car- 
dan’s rule is of no ufe in the folution of cubic equa- 
tions that have three real roots ; lince they have here 
been reduced to the other cafe in which the equation has 
but one real root, which cafe is always refolvable by 
that rule. And the firft hint of fuch reduction I re- 
ceived from Francis maseres, Efq. Cum tor Raron of 
the Exchequer, he having done me the favour to com- 
municate to me the fecond of the above four forms for 
the greateft root, in a letter of the 1 7th of July 1779; 
the inveftigation of which formula, together with thofe 
of the other three, nearly as above, I had the honour 
of fending him in a letter of the 26th of the fame 
month ; and that learned gentleman has lince commu- 
nicated to the Royal Society his faid formula, together 
with his own inveftigation of it, done in his ufual very 
accurate manner. Since that time I have feen, in the 
Memoires de l' Acad, for the year 1743, four expreflions 
limilar to the above, given by Mr. nicole for the pur- 
pofe of fumming certain terms of a binomial raifed to 
Vol. LXX. N n n any 
