76 



no distinguished celebrity as a regal city; and hence its 

 omission from the map of Ptolemy, who wrote in the century 

 preceding. 



Another fact, derived both from historic evidences and 

 existing remains, is, that, with the exception of the cahir 

 erected by the Tuatha Dedananns, all the works appear to 

 have been of earth and wood ; though forts and houses of 

 uncemented stones are found in other districts of equally 

 ancient or even earlier date. From the uniform character 

 which pervades these remains, the author concludes that 

 they are the monuments of one people ; and he thinks that 

 the fact above mentioned may help to elucidate the origin 

 of that Scotic race, which ruled in Ireland at the period of 

 their construction. 



Sir William Hamilton laid before the Academy an ac- 

 count of some investigations, in which he had recently been 

 engaged, respecting Equations of the Fifth Degree. They 

 related chiefly to three points : first, the argument of Abel 

 against the possibility of generally and algebraically resolving 

 such equations ; second, the researches of Mr. Jerrard ; and 

 third, the conceivable reduction, in a new way, of the original 

 problem to a more simple form. 



1. The argument of Abel consisted of two principal 

 parts ; one independent of the degree of the equation, and 

 the other dependent on that degree. The general principle 

 was first laid down, by him, that whatever may be the degree 

 n of any general algebraic equation, if it be possible to ex- 

 press a root of that equation, in terms of the coefficients, by 

 any finite combination of rational functions, and of radicals 

 with prime exponents, then every radical in such an expres- 

 sion, when reduced to its most simple form, must be equal 

 to a rational (though not a symmetric) function of the n roots 

 of the original equation ; and must, when considered as such 

 a function, have exactly as many values, arising from the 



