SERIES. 



, ^ being equal to the co-efficient of the fecond term with its 



to be equal to — / •^' ' -^ ; the fluent being taken under fign changed, we have 

 9-^ i + x 



the fame reftriftioa as before. And in nearly the fame way 

 he finds the fum of n terms of the former to be 



+ Z.II- + 2"" 4- &c., 



I 



I I 



"'" Zin ' 'Ziin. + 



&c. = --; 



and the latter, 

 2 



tter, 



,(■/>'. /^ ' + '".7 



= _ J f X '> X _ f X n xK. 



9 l->^ r+T- -^ I + 1 J 



denoting by z", 2"', s"", &c., or — , -^,, -^, &c. the 



fucceffive roots of the above equation. 



But we know that the values of x, anfwering to the cafe 

 of fin. X = o, are rr, 2 ~, 377, 4-, &c. ; tt denoting the 

 femi-circumference : fubilituting, therefore, thefe fucceffive 

 values of .v, we have 



In a fimilar manner, M. Lorgna finds for the infinite 

 fum of 



I I 



+ -r-i + — 

 3"'=- 4'^ 



+ 



+ 



(P + q)m - (p + 2q)m' if +39)'"' 



S = — / JLIJ! ; and the fum of n terms, 

 ?«^ m±x 



&c. 



+ &c. = 



-if 



and Z 



("'" ~ -y") " ' ■* '. when the figns are all plus ; 

 m"( m — *) 



(m — X ") X ^ x ^ when alternately plut 

 m^'im + x) 



and minus. 



For a farther illuflration of this method, we refer the 

 reader to Clarke's tranflation of Lorgna's treatife, " De 

 Seriebus Convergentibus," 4to. 1779. 



II. Circular Series. — We have ftated, when illuftrating 

 the methods of fummation employed by the BernouUis, 

 that James, although he had difcovered feveral curious pro- 

 perties of the feries, 



I 1,1 I 



— + + — +&c. 



had not been able to find its fum ; but this his brother 

 John afterwards effefted, and the folution of it is publiftied 

 in the 4th volume of his " Opera Omnia." Bernoulli 

 found this fum to depend upon the reftification of the 



or— + r^ + — + — 

 I- 2' 3' 4' 



Landen's method depends upon exaAly the fame prin- 

 ciples ; but he has rendered it more general, and exhibits 

 feveral very remarkable feries of this kind. He firft de- 

 duces the formula; for expreffing the fums of the feveral 

 powers of the roots, a, b, c, &c. of any equation 



X" + Ax"-' + Bx"-' + Cx"-^ 4- &c. = o; 

 viz. ifS'= a -^ b Jr c Jr &c. 

 S" = a" + b' + c- + &c. 

 S'" = a' ^ i' -f c'+ &.C. 



then S' = - A 



S" = - 2 B - A S' 

 S'" = - 3 C - B S' - A S" 

 S'- = - 4D - C S' - B S" - A S'" 

 &c. &c. 



Then from the two feries for tlie fine and coCne of any 

 arc x, viz. 



fin. X = X — 



+ 



3-+-5 



cof. 



X' 



— + 



2 



3-4 



.6 



••7 

 -f &c. 



+ &c 



circle, (hewing that it is equal to one-fixth of the fquare of he derives the fum of their roots, when fin. x = o, and 



the femi-circuniference of a circle, whofe radius = i. This 

 refult he drew from the known feries, which exprefles the 

 fine of an arc in terms of the arc, viz. 



fin. X = 



+ 



+ &C.; 



2.3 2.3.4.5 2-3---7 

 whicli, when fin. .v = o, becomes, after dividing by x, 



x' -)- &c. 



cof. X = o ; and then, from the preceding formula for the 

 fums of the fquares, cubes, &c. of the roots of an equation, 

 draws the values of the feveral powers of thofe quantities. 

 Thus in the feries for the cofine, when cof. .t- = o, we 



have for the feveral roots, (denoting the quadrant or — 



bye,) 



0=1 X- -t 



2.3 2 



3-4-5 



2.3, 



Or writing x = 



+ 



I I 



3^ '^ Tr 



+ 



I 1 





0=1 — 



7?^- 



3-4-5^ 



,72° 



+ &c. 



Again, multiplying by 2'", 

 I - - 4- 



2'"-* - &c. 



2.3 2.3.4.5 



Now the fum of the roots of every equation of this form 



I I 



3^ 5r 



there being no fecond term, but the fum of thefe fquared, 



I 1 1 I 1 



II I 



V "*" 3'f'- 



5^ i''^ 



4- 

 4- 



> 



t<C'~ J 



