130 PHILOSOPHICAL TRANSACTIONS. [aNNO 1751. 



Also V =: mx -{• n(i -{- py -{- &c. 



H — mx^ + «P^ + py" + &c. 

 &c. 



And consequently (1 + ")"" X (1 + ^-)" X (I + j)** X (1 + J)' &c. =A + 

 " _i- - 4- -, &c. where a := 1, b := pa, c = — -" — &c. as before. Which equa- 

 tion or theorem answers in case of a descending series. 



Carol. 3. — Hence, if each of the quantities m, n, p, &c. be taken equal to 

 unity, and their number be denoted by v; then will 



(l+l)X(l+f)X (1 +J)X(l+^)&c. be = A + J + l + ?&c. 

 Which equation, multiplied by z", gives (z + «) X (z + P) X (z + y) X (z-{-S) 



&C. = AZ- + BZ"- + CZ"-' + DZ"-' &C. 



Whence it appears, that (z — a) X (z — fS) X (z — y) X (z — i) &c. is = 

 Az — Bz'""' + cz'"-« — Dz'"~' &c. Where a = 1, b = pa, c = 



llJZ^*^ D = ^^~*^3 ^ ^ ^> &c. as before ; p being in this case = sum of all the 

 quantities «, P, y, S, &c. a = the sum of all their squares ; r = the sum of 

 their cubes, &c. &c. 



Carol. A. — Since a, P, y, S, &c. are the roots of the equation, z" — bz*"' + 

 ex"-* — DS""', &c. = ; it follows, that, if b, c, d, e, &c. be given , the sum 

 of those roots (p) ; the sum of their squares (q), and the sum of their cubes (r) 

 &c. will also be given from the foregoing equations : whence will be had 



p = B 



a = + PB — 2c 



R = — PC + OB + 3d 



S = -|- PD — GC + RB — 4e 



T = — PE + an — RC + SB + 5f 

 &c. &c. 



where the law of continuation is obvious. 



These values are the same with those given (without demonstration) by Sir 

 Isaac Newton, in his Universal Arithmetic, for finding when some of the roots 

 of an equation are impossible. 

 Pfob. 1. — To find a series expressing the value of 



(1 + ?)- X (1 + ~Y X (1 + '-)' X (1 + jV, &c. 



By putting u = (1 + ^)"', w = (1 + ^)", &c.; and proceeding as in the last 

 problem ; there will be had 

 - = !^ X (1 - - 4- "' - 4 &c.) 



