366 Mr. Herapath on new Demonstrations [May, 
Therefore it is evident that the multiplicators by which the 
succeeding coefficients are generated out of the preceding are 
- n—O n—-1 —2 a ] 
the terms of this series —-, =-, > <= Tame“, and con- 
sequently the general theorem for raising a binomial to any 
whole positive power is a* + na"""b +n. = vo +n 
n—-1 n—-2 , ape n—1 n—2 1 t 
qe egret b 6) We Meri me telslalat te 
any thing of the law of the exponents of a, 6, because it is 
too obvious to require elucidation. 
In this way would the invention of the theorem have been much 
more natural and easy, at least for whole positive powers, than 
the method of interpolations followed by Newton; and it being 
obtained for this case, the others are easily deduced from it. 
Thus for the case of 
Whole Negative Powers, 
We have A = (a+ b>" = : 
I do not say 
(a +b)" 
1 eal 
—_—-—; = by common division 
atna'b+n : a? & + &e. 
—n —n—1 en —n—-l —n—2 f2 =n} areiks Wes n—2 n= 3 
a na b—n 8 he a ae 
j?— .... &c. which shows that the general theorem for positive 
exponents, being whole numbers, is equally true for negative. 
The same principle discovers the form of the theorem when 
the exponent of the power is fractional. 
For if the terms of the fraction be m, n, and we have 4 + y = 
(a + by"; by involution we get (vW + y)"= (@ + Ob); and 
(a+d)” os) 
therefore zr + y = Gus (a + 6)” x (@+y) 
1 y ny a at, fee 
= a” (em: —n—1 py +n-—1 19 ai n-—1.9°73 
ys 
Ss -h gieaceg enik) 
=z 
1 y a 
ss —— ¥ 
+mar-*b (Sa — malta l.3: F duayede) 
z r 2°°s 
_™m ae ar y 
a a (sa wat Bee) 
m—1l m—2 J 
+ Mm a Le a2 (a...) 
+ Fe. oe ctl Game ines eee ata Nanas Sleties, doataow ree ee 
And since (a + y)" = (a+ 6)", if we expand each side and 
