VOL. LXXIX.] PHILOSOPHICAL TRANSACTIONS. 565 



common divisor be a — a, it will once only admit of 3, 4, 5, . . r equal roots ; 

 if it be a quadratic, then it will twice admit of those equal roots ; and so on. 



4. If the roots of the equation of the least dimensions be substituted for a in 

 the remaining equations, and each of the resulting values of the same equation 

 be multiplied into each other, there will result the r — 1 equations of condition : 

 and the same may be deduced also from the several equations conjointly. The 

 equations of conditions found by the first method, if the divisions were not 

 properly instituted, may admit of more rational divisors than necessary, of which 

 some are the equations of conditions required. 



^ 2. — ]. In the year 1776, I published in the Meditationes Analyticae a new 

 method of differences for the resolution of the following problem. Given the 

 sums of a swiftly converging series ax + hx'^ + cx^ + d!r* -|- &c., when the 

 values of x are respectively tt, g», ?, &c. ; to find the sum of the series when x is 

 T, that is, given Stt = ott + 1^"^ + c-rr^ -f dyr* + &c., s^ = a^ + ^e^ -f- Cf^ + &c., 

 S(r = ao- + Zjo-^ + co-^ -|- &c. &c. ; to find St = ar + h-r'^ + cr^ + &c. 



To resolve this problem I multiplied the quan- ^^^ ^ ^^^^ + ^^j^ + &c. 

 tities, stt, s^. So-, &c. respectively into un- /s^a + li^^b + life + &c. 

 known co-ef^cients «, (3, y, &c. and there re- yo-a + v<r*6 + v<r^c + &c. 

 suited as in the margin ; and then made the ^^' ^^' ^^' 



sum of each of the terms respectively equal to its correspondent term of the 

 quantity ra + r^b + t^c + &c., and consequently a7r -|- (3^ -f- y<r + &c. = t, 

 ^^2 4- j3^2 ^ ^^z _^ ^^^ _ ^2^ ^^3 _j_ ^z _|. ^g:3 _j_ ^^^ __ ^3^ ^^ J assumed as 



many equations of this kind as there were given values tt, ^, a-, &c. of x ; and 

 consequently as many equations resulted as unknown quantities a, (3, y, &c. ; 

 whence, by the common resolution of simple equations, or more easily from 

 differences, can be found the unknown quantities a, j3, <y, &c., and thence the 

 equation sought a X stt + j3 x s^ -f y X so- + &c. = St nearly. 



2. In the Meditationes ar^e assumed for -rr, ^, o-, &c. the quantities p, 1p, 3/>, 

 4p^ . . . ;^ _ 2p, n — \p, and np for t; which, if substituted for their values in 

 the preceding equations, will give x -\- 2^ -{■ 3y ■\- AS -{• &c. = n, a + 4p + Qy 

 + l6^ + &c. = n\ « + 8(3 + 27y + &c. = 7^^ « + 16(3 + 81y + &c. = w* ; 

 and if the sums of the series ax + hx'^ + cx^ + &c. which respectively corres- 

 pond to the values />, 2/>, 3/?, ... n — Ip of a? be si, s2, s3, s4, . . . s(7z — l), 

 and the sum of the series ax -}- bx^ -f cx^ + &c. which corresponds to n value 



/., 1 Ml /\ »— 1/ rt\i n — ln — 2 



of X be sw ; then will sn = ns(n — 1) — n . -^~ s[n — 2) -\- n . —^ . —— 

 s{n — 3) . . , + nsl nearly, which equation is given in the above-mentioned 

 book. 



3. The logarithm from the number, the arc from the sine, &c. are found by 

 serieses of the formula 'ax + bx^ -f cx^ + &c. ; and consequently this equation 

 is applicable to them. 



