1819.] of the Binomial Theorem. 365 
proof simple, direct, and complete. If it has any other merit, it 
‘1s to the best of my knowledge that of novelty. 
Knowle-hill House. J. HERAPATH. 
—— 
Of the Invention of a Theorem for raising a Binomial to any 
whole positive Power. 
Let a + b be any binomial, then if we take the successive 
powers of it by actual multiplication, there will result, r 
Power. 
fl. a+ 5 or the coefficients alone willbe 1, 1. 
2 at+2a b+ be 1,2, 11. 
8. a@4+3a%b+ 3a B+ 38 ie eg Sag 
4. at+4a3b+ 6+ 4a By 86h 1,4 6; 4, ¥ 
5. @&+5atb+10 a3 62410 068+ 5a bt+ BS 1,5, 10,10, 5, f. 
6. a§ +6 a5 b+15 at 67+-20 a? 63415 a? 044+64 55455 1, 6, 15, 20, 15, 6, 2. 
from which it is manifest, as well as from a consideration of the 
process of multiplication by which those coefficients are pro- 
duced, that the coefficient of any term is equal to the sum of the 
coeflicients of the corresponding and preceding terms of the next 
lower power; as, for instance, in the sixth power, 1 = 1+, 
6=5+4+1,15=10+4 5, &c. Therefore the coefficient of the 
second term of any whole positive power 7, must be x; that of 
the first term being unity. 
Again, if we divide the coefficients of the third terms by those 
of the second, we shall have 
Coefficients of second terms or 
exponents of the powers... Hh BBs An Baan Tocnbe eth + 7 
Quotients of third by second sip n=l 
‘ ApB Bs By Sande — 
efficients. ...... Penis ah i ae OPER 20 ee a Oye oe ee 
Whence am is the general multiplicator by which the third co- 
efficient is produced from the second. Consequently the power 
being n, the coefficient of the third term is x x >: 
And generally, by following the same method of dividing the 
succeeding by the preceding coefficients, we shall have 
Indices or coeff. of 2d terms 1, 2, 3, 4, 5, 6,7, 8,9.. nor 
3 
im | 
_ 
Quotients of 3d by 2d coeff. 4, 2, 3, 4, $, 8, T, See ee ee He 
4th by 3d coeff. 4 wy ty by oy Tv at os + 
5th by 4th coeff. . ay ie ty See oe ia meee — 
mthby (m—1)th +e —2., 8 t(m= 8) 
m—l? m—l? m—V? ** m=) 
n—m+ 2 
m—l 
