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



namely, the quantity b in a/ ; p^ and p^ in a/' ; and, in the radicals a/", aj", 

 twelve other arbitrary quantities, introduced by the functions b'', bj', which lat- 

 ter functions may be thus developed, 



*/' = r„.o + r„.. a/ + (r,,, + r,,, a/) a " + (r,.„ + r,.. a/) a/'^ 

 *;' = r\, + r\, a/ + (r „„ + r\,. <) a," + (r\, + r\., <) a/'^ 



In the earlier articles of this Essay, these fifteen arbitrary quantities had the fol- 

 lowing particular values, 



^~ 1152 ' Po~ 12' ^> — "' . 



^0,0 ~ i 5 ^0,1 — ^1,0 ~ ^1, 1 — '^1,0 — ^1,1 — ^l 



^0,0—55 ^0,1 "" '"l.O — '"l,! ""'"'.0 ^2,1 — *^' 



Apparent diflFerences between two systems of expressions of the third order, for 

 the four roots of a biquadratic equation, may also arise from differences in the 

 arrangement of those four roots. 



Analogous reasonings, the details of which will easily suggest themselves to 

 those who have studied the foregoing discussion, show that if we retain only one 

 radical of the third order o/", but introduce a radical of the fourth order a/^, for 

 the purpose of obtaining the only other sort of irrational and irreducible expres- 

 sion, 5^"*^ = b , which can represent a root of the same general biquadratic 

 equation, we must then suppose this new radical a/^ to be a square-root, of the 

 form 



a - = p'" (x,-x,) = 1/^'- (_ !^ + 12^3+ ^) ; 



p'" being a function of the third or of a lower order, which in the earlier articles 

 of this Essay had the particular value ^ ; while t?, has the meaning recently 

 assigned, and e^, e^ have those which were stated in the second article ; we must 

 also employ the expressions 



_ r /// , r <// 7f _ zifi 4. ii J- ^^ 

 *2— "o "1 "'I — 4 T^ 4 2p"" 



