'RACT 12.

OF THE BINOMIAL THEOREM.

229

fractional, positive or negative, rational or surd. When theexponent is integral, the denominator n is equal to 1, andthe quantity then in this form (p -j- pq)’", denotes a binomialto be raised to some power ; the series for which was fullydetermined before Newton’s time, as will be shown in thecourse of the 19th Tract of this volume. When the ex-ponent is fractional, m and n may be any quantities what-ever, m denoting the index of some power to which thebinomial is to be raised, and n the index of the root to beextracted of that power: and to this case it was first extendedand applied by Newton. When the exponent is negative,the reciprocal of the same quantity is meant; as

1

(p+pu) . is equal to «•

(p + pq)«

3. Now when the radical binomial is expanded in an equi-valent series, it is asserted that it will be in this general

ft ft. m -

form, namely (p + pq)" or f* x (1 + q)"~ =

— mm m—n in m—n m —2 n

p» x 1 + -Q + - . ~ . -aT . -^-a 3 + &c),

™ m m — n m — 2 // in —3 n

or P« X 1 + —AG + bq + ^ CQ \- —jy- DQ + &c.

where the law of the progression is visible, and the quanti-ties p, m , n, a, include their signs -|- or —, the terms of theseries being all positive when a is positive, and alternatelypositive and negative when a is negative, independent how-ever of the effect of the coefficients made up of m and n :also a, b, c, d, &c, in the latter form, denote each precedingterm. This latter form is the easier in practice, when wewant to collect the sum of the terms of a series; but theformer is the fitter for showing the law of the progressionof the terms.

4. The truth of this series was not demonstrated by-New-ton, but only inferred by way of induction. Since his timehowever, several attempts have been made to demonstrate it,with various success, and in various ways; of which however