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VOL. L.] PHILOSOPHICAL TRANSACTIONS. 279 



Here then every 3d term being required to be taken, let the series (a -f da^ 

 + gx^ &c.) whose value is sought, be conceived to be composed of 3 others. 

 X (a^bXpoc-\- c X p'x' -\- d X p^x^ + e X j&V &c.) 

 X{a^bXqcc-\-cX q'x' -\- d X q^oc^ -^ e X ^V &c.) 

 A,x\a-^ h Xrx-\- cX rV -\- d X rV + e X rV &c.) 

 having all the same form, and the same coefficients with the series first proposed, 

 and where the converging quantities px^ qx, rx, are also in a determinate 

 (though yet unknown) ratio to the original converging quantity x. Now in 

 order to determine the quantities of these ratios, or the values of p, q, and r, 

 let the terms containing the same powers of x, in the two equal values, be 

 equated in the common way. 



So shall, And consequently, 



^b Xpx -\-^h X qoc ■\-^b X rx =0 jb+^_|_r=0 



^ c X jf^V + ic X ^V + ^c X rV = p-^ + ^2 _|_ ^2 _ Q 



^dX p^cc^ + -hdX 7V + -i- c? X r V = cb^ p^ -\- q^ -\- r^ = 3 



4- e X pV + 4- e X q^x* + i e X /V = 0, &c. p^ ^ g^ ^ r^ — 0, &c. 

 Now make p^ = 1 , 9^ = 1 , and ?^ =■ 1 ; that is, let p, q, and r, be the three 

 roots of the cubic equation z^ = 1, or z^ — 1 = O: then, seeing both the 2d 

 and 3d terms of this equation are wanting, not only the sum of all the roots 

 {p + q -{- r) but the sum of all their squares {p"^ + ^^ + r^) will vanish, or be 

 equal to nothing, by common algebra, as they ought to fulfil the conditions of 

 the first two equations. Also, since p^ = l, ^^ = 1, and r^ = 1, it is also evi- 

 dent, that p'-\-q'-\-r' {=zp-^q + r) = 0, p' -\- q' + r' {= p' + q^ -\- r^) 

 = 0, p^ + q^ + r^ {=P^ + 9^ -\- ^'^) = ^- Which equations being, in effect, 

 nothing more than the first 3 repeated, the values of p, q, r, above assigned, 

 equally fulfil the conditions of these also : so that the series arising from the ad- 

 dition of 3 assumed ones will agree, in every term, with that whose sum is re- 

 quired: but those series, of which the quantity in question is composed, having 

 all of them the same form and the same coefficients with the original series a 4- 

 bx -J- cx^ -\- dx^ &c. (= s), their sums will therefore be truly obtained, by substi- 

 tuting p.v, qx, and rx, successively, for x, in the given value of s. And, by 

 the very same reasoning, and the process above laid down, it is evident, that, if 

 every wth term, instead of every 3d term, of the given series be taken, the 

 values of p, q, r, s, &c. will then be the roots of the equation z" — 1 =0;* 

 and that the sum of all the terms so taken, will be truly obtained by substituting 



360° 360'' 360° 

 * If «, /S, y, ^, &c. be supposed to represent the co-sines of the angles , 2 x • , 3 x , 



^ H 21 



&c. the radius being unity j then the roots of the equation z» — 1 = (expressing the several values 

 of p, q, r, s, &c.) will be truly defined by 1, « -f Van _ l, « — Vxx — i, /3 4. aZ/S/S _ 1^ /3 _ 

 'v^/3/3 — 1, &c. The demonstration of this will be given farther on.— <Orig. 



