1819.] of the Binomial Theorem. 367 
™ 
equate the first terms, there willresult r = a" ..... aT ©. 
m xh m m—1 «i? m xb mm—1 xb? 
een 2, ita oak te es oN oe 
n—ly? eh mn—l xb? 
— oo, +7 ——.. (3) 
Shae n 2 a 
Now since «x + y= (a + b)*, it is evident that the terms of 
equation 1, properly taken, ought to give the development of 
(a + 6)"; and this will be the case, if we collect the terms in 
the vertical order of their collocation, and make the necessary 
substitutions from equations 2 and 3. 
The first term gives ss = a"; and the second term, without 
x 
its coefficient, will be a" =; and thus it may easily be seen 
that the powers with a fractional positive exponent follow the 
same law as those whose exponents are whole numbers. There- 
fore, for the sake of brevity, we shall omit the powers of a and 
4, and consider the coefficients only. 
By substituting from equation 3, the coefficient of the second 
n—Il.m 
eet 
term becomes — +m=™, 
n 
In like manner the coefficient of the third term, by taking the 
coefficients of the three next vertical terms, is equal to n = 1 . 
es mn le ee at here woliave 
n 2 n 2 
taken no notice of the coefficient of the second term _ 
—leih ee) 2 ] ee et ae aa 
= — “— of the value of y, which was neg- 
a 
lected in the former substitution for the coefficient of the second 
term. If, therefore, we add this coefficient, multiplied by — 2 —.1 
. y.. - mm m—n 
the coefficient of — in equation 1, to > . —j— found from sum- 
x re as — 
ming the three vertical terms, we shall have = ’ at for the en- 
tire coefficient of the third term of the development of (a+)*3 
that is, the same as would be given by the general theorem for 
whole positive powers by changing 7 into =. And by following 
the same course we shall discover a like coincidence in the 
coefficients of the fourth and other terms... Whence the theorem 
is true generally for positive fractions ; and that it is equally so 
for negative may be shown by common division ; therefore, it is 
universally true for all whole and simple fractional numbers. 
