Involution of Polynomials. 241 
terms containing these letters SPaPeane from the erantion, and 
we have a+b =a"+na™"b +n zt aft y HS" 
a"~°63+&c, in which n, »—1, n—2, rt arise from — decrease 
of the powers of a, and the depoeiisnhere from the increase of 
the powers of 6. This is the well known Binomial Theorem. 
Put a+b"=a"-+na ~'6 + Aa*~*b? +Ba"-°b34+Ca"-4b44 &c, 
If a third term c is introduced, we shall have the following ad- 
ditional terms: 
2Aa"-2be+-3Ba"-2b2c+4Ca"-*b3ce+- 5Da"— 5 b4c+&c. 
If a fourth term d is now introduced, we shall have again as 
additional terms : 
2:3Ba"-*bed+3-4Ca"-*b2cd +-4:5Da"~*b*cd+&e. 
If a fifth term e, we must again add, 
2:3:4Ca"- 4 bede+3°4:5Da"- *b*cde+&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‘h*cd, a*b‘cd, ab?c*d, and abe*d*,) and having like coeffigients ; 
but only one of each of these terms has been given ; this being 
sufficient to indicate the magnitude of all the coefficients. 
When 7 is large and the number of terms in the root is small, 
it is most convenient to find the coefficients of a binomial, and 
afterwards obtain from these the additional coefficients for the 
other terms, as shown in the last process. In many cases, how- 
ever, it is better to find the higher coefficients first. 
Example 1. Find the coefficients of a+b+c*. Here we have 
P=3x2=6, 3 = 3, and the form of the power is atbtc= 
i voces 
Er. 2. Find the coefficients of ee” 
4 
reas =4x3x2=- - 
ae se = ee BB a* be 
cou 2 ee 
— - ee | a‘ 
Hence a}b+c+d‘= =a*+-4a*b-+-6a"b* + 12a" be-+ Bdabad + 8 
Vol. x1, No. 2.—Jan.-March, 1 
