22 S OF THE BINOMIAL THEOREM. TRACT 12.

26. For equations of higher dimensions, as the 5th, the 6th,the 7th, &c, vve might, in imitation of this last method, com-bine other forms of quantities together. Thus, for the 5thpower, we might compare it either with (x — n) 4 x (.r — b),or with (y — a) 3 x (.r — b) 1 2 , or with (x — a) 3 x (x — b) x(x — c), or with (.r— a) 2 x (.r— b) 2 X {x — c ). And so forthe other powers.

TRACT XII.

OF THE BINOMIAL THEOREM- WITH A DEMONSTRATIONOF THE TRUTH OF IT IN THE GENERAL CASE OF FRAC-TIONAL EXPONENTS.

1. It is well known that this celebrated theorem is calledbinomial, because it contains a proposition of a quantity con-sisting of two terms, as a radix, to be expanded in a series ofequal value. It is also called emphatically the Newtoniantheorem, or Newton’s binomial theorem, because he has com-monly been reputed the author of it, as he was indeed for thecase of fractional exponents, which is the most general ofall, and includes all the other particular cases, of powers, ordivisions, &c.

2. The binomial, as proposed in its general form, was, by

———— m

Newton, thus expressed p + pa n ; where p is the first termof the binomial, a the quotient of the second term dividedby the first, and consequently pa is the second term itself;or pa may represent all the terms of a multinomial, after thefirst term, and consequently a the quotient of all those terms,except the first term, divided by that first term, and may beeither positive or negative ; also — represents the exponentof the binomial, and may denote any quantity, integral or