118 British Association. 



equation of the fifth degree would be resolved, if, contrary to the theory 

 of Abel, it were possible to discover, as Mr. Jerrard and others have 

 sought to do, a reduction of that general equation to the binomial 

 form, or to the extraction of a fifth root of an expression in general 

 imaginary? And he conceived that the propriety of considering such 

 extraction as an admitted instrument of calculation in elementary 

 algebra, is ultimately founded on this : that the two real equations, 



x^ — lQofi y^ -{■ 5 xy*= a, 

 bx^y — XOx'^y^ -\-y^ = b, 



into which the imaginary equation 



resolves itself, may be transformed into two others which are of the 



forms 



. 5r— 10t3 + r^ 

 f ^ = r, and — — — = t, 



so that each of these two new equations expresses one given real 

 number as a known rational function of one sought real number. But 

 notwithstanding the interest which attaches to these two particular 

 forms of rational functions, and generally to the analogous forms 

 which present themselves in separating the real and imaginary parts 

 of a radical of the nth degree. Sir William Hamilton does not conceive 

 that they both possess so eminent a prerogative of simplicity as to 

 entitle the inverses of them alone to be admitted among the instru- 

 ments of elementary algebra, to the exclusion of the inverses of all 

 other real and rational functions of single real variables. And he 

 thinks, that since Mr. Jerrard has succeeded in reducing the general 

 equation of the fifth degree with five imaginary coefficients to the tri- 

 nomial form above described, which resolves itself into the two real 

 equations following, 



x''> — 1 j:3 y2 _|_ 5 jp ^4 _|_ ^ _« Q^ 



5 x^ y — \0 x'^y^ ^ y'^ -\- y = by 



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

 improvement of this part of algebra, to inquire whether the depend- 

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

 the two real numbers a and 6, cannot be expressed by the help of the 

 real inverses of some new real and rational, or even transcendental 

 functions of single real variables j 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. 



Mr. Peacock observed that the Section were scarcely aware of, 

 and could not be too strongly impressed with the value of an at- 

 tempt like that of Sir W. Hamilton to render this celebrated argu- 



