
1799] 
from it, 7.é@. more new equations have 
been refolved than by any other perfon. In 
the faid edition of 1762 was given a rule 
for finding impoffible roots of a given 
equation from an equation, of which the 
roots are the f{quares of the roots of a 
given equation; a fimilar rule has been 
jince publifhed in the Peterfburgh acts ; 
another rule for finding impoffible roots 
was alfo given from finding an equation 
of which the roots are the {quares of the- 
differences of the roots of the given equa- 
tion, by which, from the change of figns 
are always difcovered when the roots of 
the given equation are all poflible or not ; 
and from the laft term of the refulting 
equation, being either affirmative or 
negative, is difcovered whether the num- 
ber of impoflible roots is either 2, 6, 10, 
&c. or 0, 4, 8, &c.; this was publithed 
in the firft edition of the Algebra, 1762, 
and in the philofophical tranfactions for the 
year 1764, and has been fince publithed by 
fome of the greateft mathematicians. 
In the fame paper contained in the 
philofophical tranfaétions for the year 
1764, was introduced a new principle 
for finding whether the area of an alge- 
braical curve can be expreft in finite al- 
gebraical terms, by afluming an alge- 
braical equation, which neceffarily ex- 
prefles the algebraical relation between 
the area and the abfcifs, when they can 
be expreft in finite terms ; and afterwards 
I publifhed, in the Med. Analyt. therefo- 
lution of a mere general problem on 
the fame principle: Mr. Condorcet fince 
did the fame for fome more algebraical 
equations on the fame principle above 
mentioned firlt difcovered by me. 
Mr. Condorcet did me the honour to 
fend me his book on the probabilities 
of juries differently inftituted ; it contains 
many very fenfible reflections on political, 
as well as mathematical matters; I have 
not the book in my pofleflion, and I only 
fpezk froma faint memory ; it contains 
principally the application of the binomial 
or trinomial feries 
n n D-I ox 
a+o=atna btn Za 
Aa—m m n EE Rae | n n-I pa 
Pa b+). atbivmatnua % b-tc 
+ &c. to the above mentioned decifions, 
fome other problems on decitions are ac- 
ded : Mr. De Moivre and others introduced 
the binomial tor fimilar purpoles ; they 
affume @ and } for the probabilities of an 
event happening, and for its failing re- 
{peStively, and conclude that the chance 
of its happening z—m times, and tailing 
mimes in # trials will be Pa™™e™ 
(Fh 
n 

oe Pt n 
b+ Sc.—a... + 
Original Letter of the late Dr. Waring. 
307 
Mr. Condorcet affumes @ for the 
cliance of a perfons voting truly, 4 for 
the chance of ‘his voting falfely, and ¢ 
for the chance of his not voting at all, 
and from thence deduces a fimilar con- 
clufion—More of thele numbers may be 
added together, and more decifions in- 
ftituted, and their probabilities made = 
t> 3, &c. OF in any given ratio, &c. to 
each other: fome inftances have been 
given by De Moivre, &c. in the general 
reafoning, but very many on trials by 
Mr. Condorcet, but to me it appears 
-very doubtful: 1. when the chance of any 
two perfons voting truly can be affumed 
equal; or the chance of any one perfon 
voting truly can be given: and therefore. 
what weight fuch calculations can have. 
In my tranflation of algebraical quan- 
tities into probable relations, an elegant 
theorem is given, viz. ath atb—x 


ey 
od--b—2.% ab —— Bre Sparen 7 4 La a—-% eo 


a—t—ixtna.a—x.. a—t—-2x% b+-x. 
Nef 
ee 

——— 
2 + 4. G—X .U—2X 2. GI — 5 Xx 
b. bx + 

SSC 2 . deme a —2 X—a —1— 1 * - Siurauaes 
J-P. a. a—x. . aemmx. b.b.—x. be 2X2. 

be te ete Bec 
Let a and 6 denote the number of things of 
different kinds, A and B contained in an 
urn or which may denote the fame thing, 
the number of chances that an event happens 
or fails; and every time that @ is drawn 
out of the urn, let the number of A’s te 
diminifhed by x, and every time that & 
is drawn out of theurn, let the number 
of B’s be diminifhed by x; then will the 
number of chances of A’s happening cr 
being drawn (77-1) times and B’s being 
drawn z—72—1x times in z trials will be 


-. @>-b—2z—Iy, 
Every thing that can be deduced on the 
former fappofition that the probabilities 
a, and 6, &c. remain the fame from the bi-« 
nomial, trinomial, &c. may with equal 
facility on the latter fuppofition be de- 
duced from this-.theorem. 
In the year 1762, a rule was given for 
our two algebraical equations of 2 and m 
dimenfions, containing two unknown 
quantities x andy, of finding the dimen- 
fions of the quantity x, when the two 
equa. 
