306 THIRD REPORT 1833. 



to express them, when those coefficients were general symbols, 

 though representing rational numbers. Such functions also 

 must equally express all the roots, in as much as they are all of 

 them equally dependent upon the coefficients for their value ; 

 and they must express likewise the values of no quantities which 

 are not roots of the equation. 



The problem, in fact, is the inverse of that for the formation 

 of the equation which is required to satisfy assigned condi- 

 tions. And as we have shown that there always exist quanti- 

 ties expressible by the ordinary signs of algebra which will fulfil 

 the conditions of any equation with rational coefficients, so like- 

 wise we might appear to be justified in concluding that there 

 must exist explicable functions of those coefficients which in all 

 cases would be competent to represent those roots. 



A very little consideration, however, would show that such a 

 conclusion was premature. In the first place, such a function 

 must be irrational, in as much as all rational functions of the 

 coefficients admit but of one value ; and they must be such ir- 

 rational functions of the coefficients as will successively insulate 

 the several roots of the equation, — for they must be equally ca- 

 pable of expressing all the roots, — and they must be capable 

 Hkewise of effecting this insulation without any reference to the 

 specific values of the symbols involved, or to the relation of the 

 values of the roots themselves ; for otherwise they could not be 

 said to represent the general solution of any equation whatever 

 of a given degree. The question which naturally presents it- 

 self, after the enumeration of such conditions, is, whether we 

 could conclude that any succession of operations which are, pro- 

 perly speaking, algebraical, would be competent to fulfil them. 



If it be further considered that those successive operations 

 must be assigned beforehand for every general equation of an 

 assigned degree ; that every one of these operations can give 

 one real value only, or at the most two ; and that the result of 

 these operations, which must embrace all the coefficients, must 

 express the n roots of the equation and those roots only ; it 

 will readily be conceded that the solution of this great pro- 

 blem is probably one which will be found to transcend the 

 powers of analysis. 



The solutions of cubic and biquadratic equations have been 

 known for nearly three centuries ; and all the attempts which 

 have hitherto been made to proceed beyond them, at least in 

 equations in which there exists no relation of dependence 

 amongst the several coefficients, and no presvuned or presuma- 

 ble relation amongst the roots, have altogether failed of success : 

 and if we consider that this great problem has been subjected to 



