respecting Equations of the Fifth Degree. 255 



imaginary coefficients, has been reduced, by Mr. Jerraed's* method, to the 

 system of the two real equations 



af" — 10 x*j/^ + 5 xy* +x — a, 

 5 x*y— 10 x^y +y + J/ = &) 



it ought, perhaps, to be now the object of those who interest themselves in the 

 improvement of this part of algebra, to inquire whether the dependence of the 

 two real numbers x and y, in these two last equations, on the two real numbers 

 a and b, cannot be expressed by the help of the real inverses of some new real 

 and rational, or even transcendental functions of single real variables ; or, (to 

 express the same thing in a practical, or in a geometrical form,) to inquire 

 whether the two sought real numbers cannot be calculated by a finite number of 

 tables of single entry, or constructed by the help of a finite number of curves : 

 although the argument of Abel excludes all hope that this can be accomplished, 

 if we confine ourselves to those particular forms of rational functions which are 

 connected with the extraction of radicals. 



It may be proper to state, that in adopting, for the convenience of others, 

 throughout this paper, the usual language of algebraists, especially respecting 

 real and imaginary quantities, the writer is not to be considered as abandoning 

 the views which he put forward in his Essay on Conjugate Functions, and on 

 Algebra as the Science of Pure Time, published in the second Part of the 

 seventeenth volume of the Transactions of the Academy : which views he still 

 hopes to develope and illustrate hereafter. 



He desires also to acknowledge, that for the opportunity of reading the 

 original argument of Abel, in the first volume of Crelle's Journal, he is 

 indebted to the kindness of his friend Mr. Lubbock ; and that his own remarks 

 were written first in private letters to that gentleman, before they were thrown 

 into the form of a communication to the Royal Irish Academy. 



* Mathematical Researches, by George B. Jerrard, Esq., A. B. ; printed by William Strong, 

 Clare-street, Bristol. 



VOL. XVIII. 2 M 



