240 Involution of Polynomials. 



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 he increased one by dividing 

 its coefficient by the index of the power to which it is to be raised. 



And conversely, the poioer of any term may be depressed one by 

 multiplying its 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-c'^de, a^c^, &c. 



Let N be the coefficient of a-c-d, and put c=m-{-n, then 

 ]^a^c^d='^a^d m-\-n^ =^2Na-c?ww+&c. Hence, N= J coeff. of 

 a^dmn=^ coefF. of a-cde. Therefore, coeff. of a'^c^d—^ coeff. 

 of a^cde. 



Again, let m be the coefficient of a^c^, and putting c=-in-\-n, 

 we have l!lia''c''=Ma^m-\-n^ =^'S^a~m^n-\-6cG. Hence, M=i 

 coeff. of a-m~n=--^ coeff. of a^c^d. 



Therefore the coeff. of a"c^=^ coeff. of a'^c^d=^-^ coeff. of 



1 P . 



a°cc?e= (,.o.o coeff, of «5c<ie= o.o.o ' A similar process apphed 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 n (the index of the power) =n 



^^^IT ?"r^ 3-21. 



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 

 w — 1, and the final quotient n. 



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. 



4. That the coefficient of any term a^b''cd= 



P ?in, — ln — 2 m-\-l 



17273 OT X 1, 2, 3 r~ 1? 2 , 3 -7~ '^^ 



that the coefficient of a"'bcd=n n — 1 w — 2 m+1. If the 



root contains only a and b, or c, d, e, &c. are each =0, then all the 



