Sir William Rowan Hamilton on Equations of the Fifth Degree. 359? 



V346 - V543 + ^' (V453 - V435) + ^ (V534 - V354) ; 

 V345 - V435 + ^ (V453 — V354) + ^ (V534 - V543) ; 



6 being still an Imaginary cube-root of unity. We find : 



V345 - V354 = 5 (y', - y'O - dy"3 ; 1 

 V534 - V435 = — 5 (y', - y',) - dy"3 ; 

 V453-V5,3 = 2dy"3; 



(119) 



(120) 



expressions which show immediately that 



V345 + V453 + V534 = V354 + V543 + V43y (121) 



and, therefore, by (c) and (d), that 



H2 = 0, 

 as was otherwise found before. Also, 



20» _ _ 1 = (0 _ 1) (20 4- 1) = - (1 - e) (e - e') ; (122) 



and, consequently, by (120), the first of the three factors (119) is equivalent to 

 the product of the two following : 



1-e, 5(Y,-y\)-^Y'\; (123) 



in which, as before, 



f = (0 — 0^) D = a/^=T57 



But, by (112) and (117), 



2 (Y'4 - y's) = Y53, — Y435 - (y^, — y,^) = 2 (y^ - Y^) + Y,,3 - Ym3, (124) 



and 



(125) 



(126) 



•^^3 — ^453 ^543 » 



so that the first factor (119) may be put under the form : 



^ (1 - 0) {10 (y,3, - Y,3,) -4- (5 - f ) (y«3 - v^3)}. 

 Besides, by (111), the three differences 



Ycde ~~ Ycedj Ycde ~~ Yedct Ycde ~~ ^dcet \^"' ) 



