a 
Tnvolution of Polynomials. 239 
hier IV. _The Invshitian of Palicsinmite sg by ¥ Ww. J. Luwih, 
Civil Engineer, Germantown, Penn. 
Ir any binomial a+b be raised by actual multiplication to the 
2d, 3d, 4th, 5th, and nth powers, we find that the first and second 
terms éf the powers are a* +-2ab, a°+-3a?b, a'-+4a*b, a> +5a%6, 
and a"+na"~'b respectively. 
Let it be required to raise to the 5th power any expression 
a+b+c+d-+e, consisting of at least five terms. 
Then considering b+ce+d-+e first as one term, then as wobde 
up of btetd+e, and subsequently regarding de as one term ; 
and retaining only the second term of the first involution, and 
the first and second of the others, we have 
atb+etd+te’ : 
=but+b+c+d‘et+&e. ( =5ate+&c.) 
=5.4a+b+e°de+&c. ( = 5-4a*de+&c.) 
=5:43a+b'cde+&e. ( =543a?cde+&c.) 
=5:4:3-2abede+&e. (=5:43-2abede+&c.) 
Hence, if P=coefficient of abcde, then wil g = coefiicient of 
PF. r \ P 
a* bed, 5.3 = coeff. of a®be, 5.34 = coell. of a*b, and 5346 — 
coeff. of a5. 
If our root had consisted of more than five terms, P would 
have represented the coefficient of the product of any five terms, 
ls 
5% the coeff. of the product of a*, and three other terms, 5:5 the 
coeff. of a® multiplied by any two dther terms, &c. 
The coefficient of a? bed is also the coefficient of abcd, abe?d, 
abed?, &c.; the coefficient of a%be is the coefficient of ab*c, 
abe*, abd*, &c.; and generally, any term can be substituted for 
a@inthe above expressions. For either of the terms 4, ¢, d, e, 
can be placed first in the root, when it will be subject to the same 
Operations as have been performed on a, and will consequently 
be substituted for it. Our remarks, therefore, in relation to the 
Powers and coefficients of @ are equally applicable to the powers 
and coefficients of all the other terms. 
