Sir William Rowan Hamilton on Equations of the Fifth Degree. 343 

 16. In this manner It is not difficult to perceive that we may write 



V345 = g + ^^ + h 



v,,3 = ^ + 0A + 0\-, I (35) 



y^ = g + evi + ei, 



and 



y,,,=g' + h' + i', 



y,,, = g' + eh' + eH', 

 y.^^gf + G'h' + ei', 



(36) 



m 



which, 



gzizg' ={— 5ay + 60 (— 5a)- j;liCD 



+ 10{(— 5a)^ + 27/BCD}e(B^+C^+D^) 



+ 5 ( - 5a) e« (b^ + C* + D^ + 6c V + 6d^b^ + 6b^c^) 



+ 10 (- 5a) t (b* + c* + d* — 3cV - 3dV - 3bV) ; 



h=kK-\-l\\ i = k'X -{- 1' k"" ; 

 h' =k\-\- Ik\ i' = k'K + I'X^ ; 



yfc = 10 {( - 5a)^ + 2i;bcd} (b^ + ee + 0V) 



+ 10 (- 5a) e (b^ + 0c* + e^D* — 3c-D^ — 3eD^B^ — S^^b^c') ; 



/ = 5 ( - 5a) (b* + ec' + eV + 6c^D* + eCDV + 60-b'c') ; 



(37) 



(38) 



(39) 



and k', I' are formed from k, I, by interchanging 6 and <?^ Hence also, by the 

 same properties of e, t}, i, which were employed in deducing these equations, we 

 have : 



hh' = kh + l\^ + kl {'rf - e^ 4- Set) ; 



1 



h'-\-h'^ = 2{?>k''-th)U-\-{k-^?,lh)k{'>f—e^-\-^ei) J^PI^tf-e^2,eif;\ 



(40) 



and «', P -{■ i'^ have corresponding expressions, obtained by accenting k and /. 

 17. If then we make 



^ = Hi + ^/H« ^' = H, — \/h2 ; 

 h^ -\- h'^ = 2H3, h^ — h" = 2\/h, ; 



t'' + i^ = 2H„ 



V'3. 



i' = 2\/H,; 



(41) 

 (42) 

 (43) 



