Sir William Rowan Hamilton on Equations of the Fifth Degree. 369 



Tsr, l', and l", having the significations already assigned ; and the values of the 

 coefficients q and r depend essentially, in general, on the choice of the root x', 

 although they can always be expressed as rational functions of that root. 



41. It does not appear to be necessary to write here the analogous calcula- 

 tions, which show that the two remaining quantities Hj and Hj, which enter into 

 the same formula (a), (b), (c), (d), are not, in general, symmetric functions of 

 the five roots of the proposed equation of the fifth degree, nor equal to each 

 other, but roots of a quadratic equation, of the same kind with that considered 

 in the last article. But it may be remarked, in illustration of this general result, 

 that for the particular equation of the fifth degree which has been marked (147) 

 we find, with the arrangement (148) of the five roots, the values: 



H3 = 2-^3-^5»(1809 — 914^), H, = 2-^ 3-^5" (1809 + 914^); (204) 

 with the arrangement (192), 



H3= 2-^ 3-* 5^(1269+ 781^), H5 = 2-^ 3-2 5^ (1269 — 781^); (205) 

 and, with the arrangement (198), 



H3 = 0, H, = 0. (206) 



The general decomposition of these quantities H3 and Hj, into factors of the fifth 

 dimension, referred to in a former article, results easily from the equations of 

 definition (42) and (43), which give : 



<2n,= {h + h'){h + eh'){h + e^h'); 1 



2h, = {i + i') (i + ei') (i + eH'). J 



And the same equations, when combined with (40) and (38), show that the 

 combinations 



H3^ — H, = A^ h", h/ — u^ = P i\ (208) 



are exact cubes of rational functions of the five roots of the equation of the fifth 

 degree, which functions are each of the tenth dimension relatively to those five 

 roots, and are symmetric relatively to four of them ; while each of these func- 

 tions, hh' and ii', decomposes itself into two factors, which are also rational func- 

 tions of the five roots, and are no higher than the fifth dimension. 



42. In the foregoing articles, we have considered only those six quantities h 



VOL. XIX. 3 B 



