(209) 



(210) 



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



which were connected with the composition of the six functions v, determined 

 by the definition (33). But if we establish the expressions, 



Tc2ed ^^ Vcde "T" — 



T<fc2c = Vcde — + — 



Tedca — ' cde — "r" 



which include the definition (33), and give, 



■y cde ^ ^ \'^2cde "7" T^cied — Tde2c — T^dcj), 

 V cde ^ ^ (Tacde — "T — )) 



v"'c<ie=i(T2cie — — + ), 



we are conducted to expressions for the squares of the three functions v', v", 

 v'", which are entirely analogous to those marked (a) and (b), and have ac- 

 cordingly been assigned under such forms by Professor Badano, involving 

 eighteen new quantities, H-, . . Hj^ ; which quantities, however, are not found to 

 be symmetric functions of the five roots of the equation of the fifth degree, 

 though they are symmetric relatively to four of them. 



43. In making the investigations which conduct to this result, it is convenient 

 to establish the following definitions, analogous to, and in combination with, that 

 marked (111) : 



4Y cde ^ Xjcde ~}~ ^csed — ^ddc ^edcif | 



4y cde ^ Xacde — "T — » 



4Y cde ^ Xjcde — — "T ' 



for thus we obtain, 



Xacte :^ Ycde -J- '^ cde'V '^ cde "V ^ cdej 

 ■ Xc3ed ^ ^cde "t~ ' 



^de2c '—■ ^cde "T~ 5 



Xe(te2 ^^ '^cde — — "l" ' 



V'ei. = (w* — w) Y'cde + («"' — «') y" dec, 

 y"cde = (w* - w) Y"cdc - («' - "•') y'dc,, 

 y"'cde =(«." + «- 2) y'"c.. _ (a,-' + 0.^ - 



(211) 



(212) 



2)y'' 



dee* 



(213) 



