200 Sir William R. Hamilton on the Argument of Abel, 



(so that p is equal either to the product 1.2.3...*, or to some submultiple of 

 that product) 5 we shall then have an identical equation of the form > 



F.- +D,F, +... + D^iF. +Dp = 0, 



in which the coefficients d„ . . . d^ are rational functions of Oj, . . . a„ ; and 



(*) 

 therefore at least one value of the radical a. must satisfy the equation 



(*)p ^ (k)p-i (*) 



«■ + i>i tti 4- • • • + »p_i «■ +»p = 0- 



But in order to this, it is necessary, for reasons already explained, that all the 



(A) 



values of the same radical a, , obtained by multiplying itself and all its subordi- 

 nate radicals of the same functional system by any powers of the corresponding 

 roots of unity, should satisfy the same equation ; and therefore that the number q 



of these values of the radical a. should not exceed the degree^ of that equation, 



or the number of the values of the rational function f^ . 



Again, since we have denoted by q the number of values of the radical, we 

 must suppose that it satisfies identically an equation of the form 



«• +E,a +... + E^_ia, +E^ = 0, 



the coefficients e„ . . . e^ being rational functions of a„ . . . o„ ; and therefore 



W 

 that at least one value of the function f. satisfies the equation 



• W ? , (A) ?-l , . (*) , 



F,. + E, . P.. + . . . + E^_, • Fj +E, = 0. 



Suppose now that the s roots x^, . . . x^ of the original equation in x, 



> »— 1 



a; + A,x -1- . . , -|- As_i x -j- As = 0, 



are really unconnected by any relation among themselves, a supposition which 

 requires that s should not be greater than n, since a,, . • . a^ are rational func- 

 tions of a,, ... a„; suppose also that a^, . . . aJ^ can be expressed, reciprocally, 



