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



24. With respect now to the factor l, which enters by its square into the 

 expression (61), and is the numerator of the fraction which is squared in the 

 form (81), we have, by (62), (63), and (57), 



L = I (15625a9 + 24375a'e + 3750a«»; 



— l6l25aV + 1500a*t + SgOOa^ef] + 7605aV 



_ 8820a^e« — 6260aV — 1290a^€'»; + I20u'r]i. + I56aerf + 8ri') 



+ l| ^ ( 15625 (a"— a'e) + 3750a«»;— 125aV + 15500a^ — 2500a^€»7 



+ 1125aV— 4500a='«_100aV— 10aV^+1240aV— 100a6?;'' + 8i7^) ; 



(82) 



and when we substitute for a, e, »;, t, their values (16), we find, after reductions, 

 a result which may be thus written : 



2«5'l = 5l' — f l" ; (83) 



if we make, for abridgment, 



l' zr 25X' + 275^y' + ( 135p^ — 350r) j/' + 2l0pqj;'* 



+ (141/— 500pr+ SS5q^)3f' + {9Sp'q-20qr)x"'-^20pq'a/—4q 



l" = 1750^^ + 2825py^ + 2100q.v"' + (1120/ + 1825r) x" \ (84) 



+ I6l5j9yy*4. (39/ + 1060pr + 500q^) x" 

 + (109p*^ + 620qr) a;'^ + 68pq^j/ + 12q\ 



With these meanings of l' and l", the quantity H4, considered as a rational func 

 tion of a/, p, q, r, may therefore be thus expressed : 



5L'-fL' 



H4 = — 2-"3-^5 



.p(. 



bx"' + Spx"" + 2qs' 



+ J' 



(85) 



p being still the quantity (79). and f being still = v' — 15. 



25. Depressing, next, as far as possible, the degrees of the powers of or', 

 by means of the equation (2) of the fifth degree which 3/ must satisfy, we 

 find : 



(86) 



in which the coefficients are thus composed : 



