Sir William Rowan Hamilton on Equations of the Fifth Degree. 371 

 Introducing also the following notations, analogous to (112), 



y' — Y'' 4- y'" y" — y" — Y^" 



* 345 *• 5 ~ ^ 55 ''435 * 5 ^ 5> 



y' — y" 4- y'" y" 



» 453 "31" 3' "^54 



^\ - ^"\, 



y' — Y^' 4- y'" y" — V^' V^" • 



'534 ^4"* 4' '354 ''4 ' 4» 



''345 '' 5r'' 5»'^435 ^ 5 '' 5» 



y" Y^" 4- Y^"" y' V^^ v"" 



'443 "si'' 3 J "^ 513 "3 " 3' 



y" Y^^' 4- Y^"' y' — Y^"" V^^" • 



*534 * 4T'' 4> "354 '^ 4 ■' 4» 



and 



345 



.\\V „\^^" 



Y^''^ -4- Y^^"' y'" v' 



^ air '^ 5'* 435 '^ 



'■ 453 *■ 3 T^ » 3» " 543 '' 3 »■ 3» 



■' 534 " 4T^'' 4> * 354 ^ 4 '' 4' 



we find, by (30), results analogous to (113) and (114), namely. 



V — 



* 345 



v' — 



' I'll — ■ 



» All "™' 





v" — 



BY^', + CY^"„ V',33 = BY-, - CV-"„ 

 By^+CY-'3, V',3 = BY-'3 - CY-"3, 

 BY^ + CY% V'3^ = BY^-CY^^"4; 



435 ^ ^ 5 



+ BY-",, v",3, : 



= CY^ 



CY-'3 + BY-"3, V",,3 



= cy\ 



and 





CY^ 



■ BY-',, 



■BY-'3, 



BY-'4; 



v"' — 



345 



v"' — 



453 



v'" 



DY-, - 5y-"'„ y'",3, = DY-', + 5y-"„ 

 DY-^ - 5y-"3, v'",,3 = DY-'3 + 5y"-'3, 

 DY--4 - 5y-"'4, v'"354 = DV^"4 + 5y--'4. 



(214) 



(215) 



(216) 



(217) 



(218) 



. (219) 



And squaring the eighteen expressions (217) (218) (219), we obtain others, for 

 the eighteen functions v'^ v"*, v'"*, which depend indeed on eighteen others of 

 the forms y, determined by the definitions (211) (214) (215) (216), but which 

 are free, by (54) and (55), from the imaginary fifth root of unity, w, except so 

 far as that root enters by means of the combination d, of which the square is = 5. 

 44. If, now, we write like Professor Badano (who uses, Indeed, as has been 

 stated already, a notation slightly different), 



3 B 2 



