240 Involution of Polynoniials. 
We see that when the powers of a are connected with the pro- 
duct of other terms, a is changed into a? by dividing the coeffi- 
cient of a by 2, a* into a* by dividing its coefficient by 3, &c. 
That is, the power of any term may be increased one by dividing 
its coefficient by the index of the power to which it is to be raised. 
And conversely, the power of any term may be depressed one by 
multiplying ws coefficient by the index of its power. 
We have not yet ascertained whether the law may be extend- 
ed to those terms in which the powers of two or more terms of 
the root are combined, as a?c7de, a*c*, &c. 
Let N be the coefficient of a*c? od, and put c=m--n, then 
Na?c?d=Na?dm+n* =2Na*dmn+&c. Hence, N= 4 coeff. of 
a?dmn=% coeff. of a*cde. Therefore, coeff. of a?c?d=4 coeff. 
of a’ cde. 
Again, let m be the coefficient of a*c*, and putting c=m+n, 
we have Ma*c*?=Ma?m-+n*°=3Ma?m?n+&c. Hence, M=$ 
coeff. of a*m?n=4% coeff. of a? 2rd. 
Therefore the coeff. of a2c?=4 coeff. of a2%c?d= es 33 coeff. of 
arele=r55 coeff. of abode= 55 A similar process applied to 
any combination of the powers of the terms of the root, will evi- 
dently show, that the coefficients are governed by the law which 
has been given. We remark then: 
1. That P, the coefficient of the product of as many terms of 
the root as there are units in m (the index of the power) =” 
n-1l n—2 3271 
2. That the coefficients of terms involving the powers of one 
letter and the product of others are obtained by dividing P by 2, 
this quotient by 3, this again by 4, &c., the last divisor being 
n—1, aud the final quotient 7. 
3. That the coefficients of terms combining powers of two or 
more terms are obtained by dividing these results by 2, 3, 4---~ 
n —2 successively. 
A. That the coefficient of any term a”b’cd= 
P nn=1n=2--->m+h and 
1, 2, 3----- mI, 2,3 ----6 r 1, 2, 3----- r 
that the conttidiant ad a "hed a1 n—2----- m-+1. Ifthe 
root contains only @ and b, orc, d, e, &c. are each =0, then all the 
