F360 Ree Mathematical Correfpondence. Su P. 
would invariably be a negative quantity only (—2). From whence, in the conclafion of his 
paper, he charges Mr. Emerfon with having committed a miftake, by imadvertently having con- 
fidered imaginary quantities abfractedly. : 
It happens, however, againft this affertion, that the proof brought to fupport it is by no 
means to the point. For though 4/—a and a/—b are imaginary quantities, it does not thence 
follow that fae is alfo one, but Phe contrary, when c 1s fuppofed greater than a. Mr. 
Garnet ought, therefore, to prove that ft «4/—b will produce ++ 4/26, independently of any 
other quantity but that (*) which was to equate their value. ; ‘ 
Suppofe theny — ja ==0, then, by Mr. Garnet’s reafoning, x = Jab ; by which 
notation I prefume he means that the quantity (4/25) under the vinctlum, is invariably -. 
For if that be denied, fuppofe the root extracted, and call it 4 7, then we have x =+ +2, 
which, I think, Mr. Garnet will himfelf allow to be nonfenfe. 
This being the cafe, lets fuppofe 5 equal to a, and then x t/a? =a fae : PR eae (cafe 1} 
For the expreffion being generally true, muft hold good in every value of af 4b; let thefe factors 
be what they may. And this proved, we have 4/—a.4/——a =-+--+|-2==x%, of certain cone 
fequence. 
Hence it wold feem that Mr. Garnet has fallen into an error, from the direétly oppofite caufe 
he has fuppofed Mr. Emerfon’s to fpiing from, viz. reafoning from equation. For, fuppofing - 
é 2 
x . . - “ 
— pte = 0, it is certain that whatever x is, it will, from the nature of adfeéted equations, 
have two equal values + 4/24, and —a/a?, differing only in the figns. Wherefore any con- 
clufion drawn from fuch premifes, proves neither for nor againft his argument, the double fign 
being an effe, the refult of a caufe wholly independent of that which arifes from the multipli- 
‘eation of the imaginary quantities. ’ 
Reaffuming, then, the equation ./——-a .4/—2==-++----a4 =, Since, as we have juft now 
roved that the double fign affixed to 2 has nothing to do with its value, as. applied to its being 
the produdt of 4/—a .4/—-a, it follows then that their value is -b a. 
Indéed, the attempt to prove the general properties of imaginary quantities, by any conclufions 
drawn from particular equations, appears (to me) equally impradticable and ebfurd. For in- 
ftance, allowing 4/—< .4/—a==—4. ‘Then fince we know that —4/2. --4/a is alfo equal 
to —a, we have, from the nature of geometrical progreifion, — 4/a:4/—a ::4/—a:--a/a. 
Now, it has been proved that each of the equal means miuft be greater than one extreme, fup- 
pofe thane /2. Then, multiplying thofe umequal quantities by —4/ a, we have @_lefs than- 
—4/—a?, which is impoffible, fince the Jaft expreffion cannot produce a value greater than a. 
Newcaftle-upon-Tyne, Iam, fir, your moft humble fervant, . 
O&. 17, 1797. | SES 

For the Monthly Magazine. 

é 
A new DEMONSTRATION oF THE RULE FOR FINDING THE SUM OF THE Power? 
oF THE Roots OF ANY EQUATION. 
[Concluded from No. XXIII] 
IV. BUT from de Moivre’s theorem for ra¥ing an infinite mulrinomial to any given power, 
it is manifett that a general expreffion for the fums of the m'> powers of a, 8, y, os & &c. 
may be eafily deduced ; for ir A be the firft term of P™,B the fecord term of P™—1,C the third 
term of P™—?, D the fourth term of P™~3,&c. A, B, C, D, &c. may be found by that 
e a * 0172 *” 
theorem, and thence, by Sect. ii, aap Bmp Map omt pm c= Abas B+ paUae x 



ws ih LI hs (2) ; 
Cry 3 Oa Gr ee fee 
Now Prpm— inp tarp t ? 
; m——1 2 
~ 2p,m—2 =” 
| +2. vi) 5 
Me Le 
vem FJ, ¢ Rae gipm—3 i 
2 4 
H—F = 
—t, ———" Zur m—2 
ae? 
= Fj yee J 
