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



o,'" and Oj'" in 6'", and a'^ in h'^, have all had the coefficients of their first 

 powers equal to unity; and consequently have been themselves expressed as 

 rational (though unsjrmmetric) functions of the roots of that equation in x, which 



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the function h satisfies ; namely, 



O2 ^^ i (•^1 ^i 'T •"'3 *4/> 



the first expression being connected with the general quadratic, the second with 

 the general cubic, and the three last with the general biquadratic equation. We 

 shall soon see that all these results are included in one more general. 



[8.] To illustrate, by a preliminary example, the reasonings to which we are 



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next to proceed, let it be supposed that any two of the terms t /„) are of the 



Pi , ■ . . 



forms 



8,1, 3, « 8, 1,3,4 I 2 3 4» 



and 



f =10 a a a a , 



1,1,J, 3 1,1, 2,3 1 J 3 "^4 ' 



in which the radicals are defined by equations such as the following 



their exponents a^', a^', a^', a^' being respectively equal to the numbers 3, 3, 5, 5. 

 We shall then have, by raising the two terms t" to suitable powers, and attending 

 to the equations of definition, the following expressions : 



// 10 / 10 /6 13 ;6 /8 1/2 II 



'2. 1. 3, 4 ^= "2, 1, 3, 4 y 1 J 2 y 3 y 4 *^1 '^a 5 



;/ 10 / 10 13 13 /4 (6 // // 



^1, 1, 2, 3 ^^^ ^1, 1, 2, 3 J I J 2 J 3 / 4 ^1 % 5 



// 6 /6 U 12 13 /4 113 II A 



^2, 1, 3, 4 = ^2, 1. 3, 4 y I y 2 7 3 fi ^3 ^4* 



