308 Original Letter of the late Dr. Waring. 
equations are reduced fo as to exterminate 
y, ifthe terms of the greateft dimenfions of y 
contained in the two equations be re(pec- 
tively 2 and m—d¢A;, by reafoning on 
the fame principles, it is finee proved, 
that the dimenfions of the equations, 
- whofe root is y, will be zm—rs, if one 
of the terms in which y has the greatett 
dimenfions in the refpective equations be 
& n—r Z 
x Y in the one equation, and 
s m—s 
xy in the other. 
Mr. BuzouT deduced this propofition 
and extended it to more (4) equations, 
involving (b) unknown quantities x, y, 
z, &c. of which the terms of the createft 
dimentions of y, &c. are correfpondent. 
From the principles mentioned above may 
be deduced whether the equations refulting 
will afcend to the dimenfions given by 
Mr. Bezout er to lefs. 
The propofitions given by Mr. Bezout 
may perhaps with equal or greater fa- 
eility bededuced from the principles pub- 
lifhed in 1762. 
In the fame book 1762, arnlets given 
from having (x) independent equations 
containing only z—2 unknown quan- 
“tities of fo reducing them, that there 
may refult (7) equations, fince called 
equations of condition. 
As alfo from one or more (7) fimple equa- 
tions having two or more (7-+-”) unknown 
quantities x,y,z, &c. of which the di- 
menfion in'each is only ene, of finding 
their integral correfpondent values in 
terms of a, 8, y, &c.; wherea, 8, y, &c. 
denote any whole numbers whatever ; 
thefe have been both (fince they were 
given by me} publifhed by others. 
Several other new propofitions, or rules 
of mine, have been fince publithed by 
foreign mathematicians, and fome by the 
Englith. 
In the above mentioned book, 1762, 
was publithed a method of finding a quan- 
tity, which multiplied into a given ir- 
rational quantity will produce a rational 
“product, or which will confequently ex- 
terminate irraticnal quantities out of a 
given equation,—this is performed from 
the roots of an equation x"-+-1-=0: ano- 
ther method in the faid book was given 
of reducing a given equation fo as to ex- 
terminate its irrational quantities; Mr. 
Bezout has fince alfo given a rule for 
exterminating irrational quantities. 
Having fufficiently fhewn that many 
mathematical inventions of mine were 
held in fome eftcem by the principal of 
the prefent 4ime, and icme of the firlt 
[ May 
mathematicians that ever exifted; for 
otherwife they would not have publithed 
the fame. It remains for me to give 
fome account of them; particularly thofe 
contained in the algebra, as it is the book 
which was firft publifhed, and has been- 
principally read. 
In the firft chapter, are delivered feveral 
elegant (as appears to me) rules, for find- 
ing the fum of any funétions of the roots 
of a given or given equations.—I may 
particularize one, as it has had the honour 
of being publifhed in the Paris Aéts, by 
Mr. Le Grange, one of the greateft 
Mathematicians that ever exifted, and 
perhaps is fuperior, in fome refpeés, to 
every rule yet publifhed in Algebra: I 
muft mention, alfo, another rule, for find- 
ing the fum of the powers of the roots of 
a given equations, in terms of the co- 
efficients of the given equation; Sir J. N. 
before found the fums of the fubfequent () 
from the fum of the preceding 7—1, »—2, 
N—3)..-%3, 2,1 powers; but my rule, 
when the feries converges, that is when 
one poffible root is much greater than any 
other, not only finds the fum of the 
powers, but alfo the fum of the roots ; 
from it has been given by me the law, 
2 
which the reverfion of a feries ya x--bx 
eae &c. obferves, and fome other 
problems, by no other method, yet dif- 
covered. Perhaps, the rule for finding 
an equation, whcfe roots are any power 
of the rcots of a given equation, may 
properly be mentioned, as by it any ir- 
rational quantities may be exterminated. 
I fhall, alfo, add the transformation of 
equations, as Mr. Le Grange prefers it 
to any other,—and laftly, the method 
given, of findine the coefficients of the 
terms of the transformed equation, from 
particular cafes, a method in thefe fub- 
jects fuperior to any other, and of great 
utility ; all of which were publifhed in_ 
the year 1762. 
The fecond chapter treats of the affir- 
mative and negative, and impoffible reots, 
and the Iimits of equations, &c. I fhalk 
particularize the rule for finding tvhether 
the two poffible roots of a biquadratic 
equation are affirmative or negative, when 
the other two are impoflible; becaufe 
Mr. Le Grange has done me the honour 
to demonftrate it; alfo, the rule for find- 
ing the number of impoflible roots, from an 
equation, of which the roots are the {quares 
of the differences of any two roots of 
a given equation, and thence deducing 
whether all the roots of a give equation 
are 
