Sir William Rowan Hamilton on Equations of the Fifth Degree. 373 



8y^^^ = ; 8y^% = 2000 ; 8y^^^', = - 2000 ; 1 



8y"^'j=— 300; 8y"'"3 = 900; 8y"^", = — 1200 ; J 



whence, by (219), 



■^v"'^ = Z; ^f^v-,3=-9+4D; Tf7V-3,= 12-4D; | 

 ^^y-,, = _3;Tf7v"',3=9 + 4D; ^-f^v'^,, = - 12-4d; J 



and the squares of these six second members are 



9, l6lq:72D, 224q:96D, (226) 



so that we have, by (220), with this arrangement of the five roots of the equa- 

 tion (147), 



H,9=2-'3-'5n97. (227) 



But with the arrangement (192), we find, by similar calculations, 



^v-3,3 = 6 + 4d ; Tf7v'"4S3 = - 9 - 4d ; yf^ v'"^ = - 3 ; j ^g) 

 Tf^v-3,= -6 + 4D;^v'"^3Zz9-4D; ^v'"3,,= +3;j 



of which the squares are 



116±48d, 161±72d, 9; (229) 



and we have now 



H,g = 2-'3-'5«iri3, (230) 



a value different from that marked (227). And, finally, with the arrangement 

 of the roots (198), we find instead of the quantities (225) or (228), the follow- 

 ing: 



ipl8-8D, ±6, 0, (231) 



of which the squares are 



644±288d, 36, 0, (232) 



and give still another value for the quantity h now under consideration, namely, 



H,9 = 2' 3-» 5' 17. (233) 



46. The twelve other expressions which have been referred to, as being ana- 

 logous to (a) and (b), are of the forms : 



v\5 = H, + '/h8 + v^Hg-f/Hio-f v^H„- \/h,2; (a') 



v'*364 = H, - /Hg -f \/h, - \/h,o + ^/h„ 4- /H,2 ; (b') 



