94. Hence, in the equation "+w=0,

On+an+3n+n+&C••• +0xn=—nw (1—w+w3 — w3 +'&c...±wk

=

-nw(±w*—1)_nw(±w*—1)

where the plus or minus sign must be used according as k is
an even or an odd No.

And, in the equation x-w=0,

On+an+an+an+ &C......• +xn=nw (1+w+w3+w3+&c...wk−1)

95. If w=1, then, in the equation x+1=0,

On+an+an+an+ &C... +%n= −n(1-1+1-1+&c. to k terms)

=-n or 0, according as k is an odd

And, in the equation x-1=0,

[or an even N°.

On+an+3n+n+ &C... +kn=n(1+1+1+1+&c. to k terms)

=kn.

In this latter case, indeed, each of the quantities, an, Tan, &C.
(by Art. 93), are equal to n.

r

96. From what has been shewn throughout this Section, it appears that the sums of the powers of the roots of any equa"tion are possible, whether the roots of that equation be "possible or impossible." This arises from the manner in which the impossible quantities destroy each other in the sums of the several powers of the impossible roots. For instance, let a, ß, y &c. a+√-b, a--b, be the roots of the equation (A) x"—pn"—gn~~rn+ &c...±w=0, then

(a+√—b)TM=aTM+mam~'~~b- m (m-1)

(ab)"a"-maTM~'√ — b —

2

m(m-1)(m-2)
2.3

24

..their sum=2a" —m (m— 1) a′′¬b+&c.

Hence

Hence the sum of the mth powers of the roots of this equation (cm) = cm +ßm+y+&c.....+2aTM —m (m—1) aTM¬3b+&c., a possible quantity; and as (by Art. 14) these impossible roots enter every equation by pairs, the same would be true whatever be the number of impossible roots.

SECT. II.

On the method of finding the sums of certain combinations of the sums of the powers of the roots of any given equation.

97. Before we proceed to find the sums of the combinations of the powers of the roots of an equation, it will be proper to explain the method of notation which will be made use of for

that purpose.

Supposing the roots of the equation (A)=0, to be a, 6, 7, , &c. then the sums of the kth, 7th, mth, (k+1)th, (k+m)th, (l+m)th, (k+l+m)th, &c. powers of those roots, will be denoted, as before, by

And,

Tky Fly Tm9 Fk+19 Ok+m, l+my Ok+i+m, &c. &c.

1. A quantity of the form

ak ßk + akk+ak Jk + Bk yk + Bk Jk + 2 * Jk + &c.

being that function of the kth powers of the roots which contains the sum of the products of the terms of taken two and two together, will be denoted by

11. The function

[F] k

ak Bk k +ak Bk gk + akyk gk + Bk zkk + &c.

σκ

which contains the sum of the products of the terms of taken three and three together, will be denoted by

III. The quantity

ak B2 + a2 ßk +ak z2 +a2yk +ßk y2+ß1zk + &c.

which contains the sum of the products arising from the mul

tiplication

tiplication of each of the terms of by each of the terms of 19 will be denoted by

Iv. A quantity of the form

a* BlgTM +a2ßkyTM +ak BT gl + aTM Bk gb + a2ßTM gt +aTM BF g + &c. which contains the sum of the products arising from the multiplication of each of the terms of σ, 19 m together, will be denoted by

v. If m=l, then this latter quantity becomes

2 (cet B2 yt +ce? Bk ge2 + ce2 ß2g2 + &c.)

a quantity, in which one of the terms of two of the terms of ; this function of coefficient, 2); will be denoted by

is combined with and σ (without its

vr. If lk, then the quantity in the last article becomes

2.3 (ak ßkyk +&c.)

=2.3 [F]

(See No. 11)

98. Now, let a*+*+*=* be multiplied together, and a2+ß2+z2=az ¶

then, a++ßk +2 +gk+l,

α

+ak B2 + a2 Bk +ak z2 + az2gt + B* ge2 + B2 gk S

And, adopting the notation of Art. 97, No. u, we have
•k++ [F] °°1=%%

i. e. ak ß2 + a2 ßk +

99. If l-k, then

aak + Bak+zak +2 (œ* B* +akyk + B*igh) = (0%)®•
Or, adopting the notation of Art. 97, No.. 1,
°2k+2 [F2] ok = (%)?

or 2 [F]=(k) — σ1k9

2

+al+m 8k + a* Bi+m + astm gik tak gltm + ßl+mg* + B* gl+m { = 0x010m0k+ 10 m +ak B2ym + a2 ßk ym + ak ßm gb +am ßk gl + a2 ßm gk +am Blyk

And, referring to the notation in Art. 97, Nos. III, IV, we

101. If ml, then (by Art. 97, N°. V), [F] becomes 2[F][F]; in this case, therefore, we have

2[F.]°x[F]°z=0x(01)3 — 20%+301-02 10%+2%+21 (T) ··• [F.]on[F] •;=** (•1)" —— 2 •x+1971-0819+20%+14

2

102. If l=k, then (by Art. 97, N°. VI), the function marked
(T) in the foregoing Article becomes 2.3 [F];
Hence 2.3[F]=(0x)3 −3240%+23*
and [F,]= (•)3—3•,*°*+2o ̧* ̧

i.e. akßkyk=

2.3

103. To facilitate the computations, the foregoing operations have been performed on the sums of the powers of three roots (α, 6, 7) only, but the values of

[F]°x; [F]; [F]°x71m; [F]ox[F]; [F]°, would have been the same, whatever had been the number of the roots; the conclusions deduced are, therefore, true for equations in general. We now proceed to give some

examples.

Let a, ß, V,

EXAMPLE I.

> be the roots of the equation
x1 − p x2+ q x2 —†x+s=0;

it is required to find the value of

Leta, B, y be the roots of the equation x3-px2+qx—r=0; it is required to find the value of

• For instance, let a* + ß* + 2k + dk + &c.=*be multiplied
and a+++d2+ &c. =σ together,

then a++++*+ !+dk+ !+&c. (x+1)? =0x01
+akß2+a2ß* +&c. ([F]oxo)

and so of the rest.