[ 759 ] 
not only the Turn of all the roots (p q -f r) but 
the fum of all their fquares ( 'p 1 + y 2 + r z ) will va- 
nifh, or be equal to nothing (by common algebra), 
as they ought, to fulfil the conditions of the two firft 
equations. Moreover, fince p l — i, q l = i, and 
r 3 = i, it is alio evident, that /> 4 + f + r4 (= P + q 
4 - r) — o, p sj rq s + r* (= p r + q 1 + ^ 2 ) = o, p 6 + 
q 5 + r 6 ( —p 3 f - q l - \- r 3 ) = 3. Which equations be- 
ing, in effedt, nothing more than the firft three re- 
peated, the values of p , y, r, above affigned, equally 
fulfil the conditions of thefe alfo : fo that the feries 
arifing from the addition of three afifumed ones will 
agree, in every term, with that whofe fum is re- 
quired : but thofe feries’ (whereof the quantity in 
queftion is compofed) having all of them the fame 
form and the fame cojficients with the original feries 
a 4- bx + cx 1 + dx l , &c. (.== S)„ their fums will 
therefore be truly obtained, by fubftituting px, y.v, 
and rx y fuceeffively, for x, in the given value of S. 
And, by the very fame reafoning, and the procefs 
above laid down, it is evident, that, if every term 
(inftead of every third term) of the given feries be 
taken, the values of p, y, r, s. See. will then be the 
roots of the equation z n — 1 = 0*; and that, the 
* If a., fi, y, S', &c. be fuppofed to represent the co-fines of the angles 
3 6o ° 2 x 4° ., 3 x &c. (the radius being unity); then the 
n ’ n n 
roots of the equation z» — i~o (exprefling the feveral values of />, q , 
S) &c.) will be truly defined by i, & -f- V — i, <* — i, 
0 _p y'Mp— I, P — V — 1, M'c. The demonflration of this 
will be given farther on. 
