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



and 



0,W^ 



"^- 1.2.3 ^+--- 



a(a-l) 

 1 — J _ 2 T--t- . . . 



and observe that these two real and rational functions (j>^ and ^^ of single real 

 quantities have always real inverses, ~' and ^ ~', at least if the operation ~' 

 be performed on a positive quantity, while the function 0~' {f) has but one 

 real and positive value, and the function 0~' {t) has a real values ; we see that 

 the determination of x and y in the equation 



X -\- V— yy - /a + V— 1 b, 



comes ultimately to the calculation of the following real functions of single real 

 variables, of which the inverse functions are rational : 



^'+y = *,"'(<'' + »■)! 



1 = *.-©; 



and to the extraction of a single real square-root, which gives 



Now, notwithstanding the importance of those two particular forms of rational 

 functions 0, and 0^ which present themselves in separating the real and imaginary 



part of the radical \/a + V^-ib, and of which the former is a power of a single 

 real variable, while the latter is the tangent of a multiple and real arc expressed 

 in terms of the single and real arc corresponding ; it may appear with reason that 

 these functions do not both possess such an eminent prerogative of simplicity as 

 to entitle the inverses of them alone to be admitted into elementary algebra, to 

 the exclusion of the inverses of all other real and rational functions of single 

 real variables. And since the general equation of the fifth degree, with real or 



