Involution of Polynomials. 239 



Art. IV. — The Involution of Polynomials ; by Wm. J. Lewis, 

 Civil Engineer, Germantown, Penn. 



If any binomial a-\-hhe raised by actual multiplication to the 

 2d, 3d, 4th, 5th, and nX\\ powers, we find that the first and second 

 termsof the powers are a2_|_2a6, a^ + ^a^b, a'^-{-Aa^b, a^-\-5a'^b, 

 and a^-l-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 6+c+c?-{-e first as one term, then as made 

 up of b-{-c-{-d-re, and subsequently regarding d-\-e as one term; 

 and retaining only the second terra of the first involution, and 

 the first and second of the others, we have 

 a + b-{-c+d + e^ 



= 5«-f-6+c+o^^e-|-&c. ( =5a''e+&c.) 



= 5. 4(7+6+7; '<^e+&c. ( =5-4a='d'e+&c.) 



= 5-A-3a + b'cde-{-&.c. ( =5-A-3a^cde+6cc.) 



= 5-4.-3-2abcdei-&c. {■■=5-A-3-2abrde-{-6cc.) 



P 



Hence, if P= coefficient of abode, then will q"= coefficient of 



P P P 



a- bed, Q7o = coeff. of a^bc, Q.o.4 ^=coeff. of a'^b, and n.o.A.K = 



coeff. of a^ 



If our root had consisted of more than five terms, P would 

 have represented the coefficient of the product of any five terms, 



P P 



o the coeff. of the product of a", and three other terms, q3 the 



coeff. of a^ multiplied by any two other terms, &c. 



The coefficient of a-bcd is also the coefficient of ab"cd, abc^d, 

 abcd^, &c. ; the coefficient of a^bc is the coefficient of ab^c, 

 abc^, abd', &c. ; and generally, any term can be substituted for 

 a in the above expressions. For either of the terms b, c, 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 a are equally applicable to the powers 

 and coefficients of all the other terms. 



