Involution of Polynomials. 241 



terms containing these letters disappear from the equation, and 



— TT" n. ,.,7, ^~1 , W-l'Al-2 



we have a + o =a -\-7ia ^o + n — n'-a^'^o^ -\-n —^ q — 



a"-353-f&c. in which n, n — 1, n — 2, &c. arise from the decrease 

 of the powers of a, and the denominators from the increase of 

 the powers of b. This is the well known Binomial Theorem. 



Pat «+ 6"= a" 4-720"- '6 + Aa"-262_}_Ba"-3^,3_|_Q^n_4j4_f.^jj^c. 



If a third term c is introduced, we shall have the following ad- 

 ditional terms : 



2Aa"-26c+3Ba"-362c+4Ca"-*63c4.5]3^n-5j4p_l_^^^ 



If a fourth term d is now introduced, we shall have again as 

 additional terms : 



If a fifth term e, we must again add, 



2-3-4Ca"-''6c(^e + 3-4-5Da"-^62c(^e+&c. 



We must not forget that there are several terms in the expres- 

 sion for the power, involving like powers of different letters, (as 

 a'^b^cd, a-b'^cd, ab-c'^d, and abc-d*,) and having like coefBoients; 

 but only one of each of these terms has been given ,' this being 

 sufficient to indicate the magnitude of all the coefficients. 



When n is large and the number of terms in the root is small, 

 it is most convenient to find the coefficients jf a binomial, and 

 afterwards obtain from these the additional coefficients for the 

 other terms, as shown in the last profess. In many cases, how- 

 ever, it is better to find the higher coefficients first. 



Example 1. Find the coefficients of a + 6 + c^. Here w-e have 



p 



P=3x2=6, o = 3, and the form of the power is a-\-b+c^ = 



a='_}-3a26c+6a6c+&c. 



Ex. 2. Find the coefficients o{ a+b+c+d\ 

 HereP=4x3x2= - - 24 

 P 



2~ 



2-2.' 



23 

 P 



12 



2-3-4' 



1 



abed 

 a^bc 



a^'b^ 

 a^b 



Hence a+6+c+rf' =a« -i-Aa^bi-ea^b^ + 12a^bc+2Aabcd+&c. 



Vol. xLii, No. 2.— Jan.-March, 1842. 31 



