Sir William Rowan Hamilton on Equations of the Fifth Degree. 331 



3. Again, if with the same meanings of ^', . .. x^', we form t by the same 

 expression as before, but suppose a to be a root of the equation 



a^ + a 4- 1 = 0, 

 then 



m = 6, n = 3, /i = -^ = 90, ^ = 30, , ^ ^. = 15; 

 8 n n{n—\) 



so that the function t will now have 90 different values, but its cube will have 

 only 30 ; and if that cube be reduced, by the equation v^ z=. 1, to the form 



^rr^^o' + ar + a^r, 



then 1'°' will be a root of an equation of the fifteenth degree, while ^ and ^" will 

 be the roots of a quadratic equation, the coefficients of this last equation being 

 rational functions of ^'°', and of the given coefficients a, &c. 



4. And if, in like manner, we consider the case 



m = 5,n = 5,fji = 120, ^ = 24, - , ^ ,, = 6, 



n n{n — \) 



so that o(f , . . x^ are the roots of a given equation of the fifth degree 



X' — KX^ -|- -Qx'^ — cr^ ■\-ttX — E = 0, 

 and 



t=x' -^ ax" + c? x"' + a^x"'-i- a'x'', 



in which a is a root of the equation 



a* -1- a' 4- a^ -j- a + 1 = 0, 



then the function t has itself 120 different values, but its fifth power has only 

 24 ; and if this fifth power be put under the form 



f = ^o' + ar 4- a" r' + aP ^" + a' ^'\ 



by the help of the equation a* = 1, then ^"^ is a root of an equation of the sixth 

 degree, of which the coefficients are rational functions of a, b, c, d, e, while 

 ^, ^", ^'", ^""^ are the roots of an equation of the fourth degree, of which the co- 

 efficients are rational functions of the same given coefficients A, &c., and of 1'°'. 



5. Lagrange has shown that these principles explain the success of the 

 known methods for resolving quadratic, cubic, and biquadratic equations ; but 



2 u 2 



