5/6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1795. 



used in the multiplication in the 3d article, be equal to each other, and conse- 

 quently each equal to a, each of the members in any coefficient will become a 

 power of a; and each term in an equation will consist of a power of a multiplied 

 into a power of x, having such a numeral coefficient prefixed as expresses the 

 number of members in the coefficient, when exhibited in the manner of the 3d 

 article. And as n expressed the number of quantities a, h, c, &c. used in the 

 multiplication, when each of these quantities is equal to a, it will denote the 

 power of the binomial x -]- a. Hence, if m denote the numeral coefficient of 

 any term of the nth power o( x -\- a, and p the exponent of a in that term, the 

 numeral coefficient of the next term will be expressed by jw X , as is evi- 



dent from the last article. 



11. It is manifest from the 3d article that x -\- a being raised to the nth power, 

 the series, without the numeral coefficients, will be x" + ax"-^ + d^x"'^ -f- 

 a^ x"'^ +, &c. and as the coefficient of the first term is 1, and of the 2d n, from 

 the general expression in the last article (x + ^0" = '^" + nax"-' -|- ra X 

 IZJ. aV- -f- n X ^— X ^— «V-' -I- &c. 



12. If the equations be generated from {x — a) X {x — b) X {x — c) X 

 {x — d), &c. the coefficients will be the same, excepting the signs, as those 

 which result from (x + a) X {x + Z') X {x + c) X (^ + d), &c. in the 3d 

 article; and as — X — gives +, but — X — X — gives — , the coefficients, 

 in equations generated from (x — o) X (x — b) X {x — c) X {x — d), &c. 

 whose members have each an even number of quantities, will have the sign -f-, 

 but coefficients whose members have each an odd number of quantities will have 

 the sign — . And lience it is evident that {x — a)" = x" — nax"~' + " X 



a'x— — n X — „— X —T— aV-' + &c. 



13. Having thus investigated the binomial theorem, as far as it relates to the 

 raising of integral powers, Mr. R. proceeds to demonstrate, by the principles of 

 multiplication, the most general case, viz. that 



_- I z 



(x -\- z)' = x' + - zx' + ; X '~Y' ^"'^"^ ~t~ ^^' This, he says, will 

 clearly appear after it has been proved that if the series x -\ — zx -{■ - X 



n n I I r , 



- — z'x' -\- &c. be multiplied by the series x -\ — zx' -|- - X —z- z?x' 



« + I "4-1 ° + 1 g+ 1 



-f- ecc. tne proauci wui oe x -j- ■ zx -\ -— X — :; — ?■ x 



-j- &c. 



