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



— 2-« ^-^{25x'^ + 75;?^* + (48/+ 45r) x'^-\- 27pqx' \ (72) 



— 2p^ + 72pr — 27q''}; 



(K'+\J=(rf—e'+3eiy=2-''3-^{625x"'+3l50p3f''+(8025p'+2250r)s" 

 + ISbOpqx" -\- (7100/ + 10350pr — 1350y') x" + 4050pV" 

 + (2004/ + 15120pV — 4050j9y' + 2025r') x'* 



+ (2592/y + 2430p^r) a:'' ^(73) 



+ (— 192/ + 6732/r — 1863pY + 6480pr" — 2430jV) x'^ 

 + (— 108pV + 3888py — U58pq') x' 

 + 4/ — 288/r + 108/^' + 5184/r^ — 3888^yV + 729*7*1 ; 



4v='V = 4t' = 2-'" 3-« { lOOOy" + 330qpa;'«' + 2700^0/" 



+ (3930p'+ 3600r) j;'«+ 5940pya;"+ ( 1991;)'+ 7920pr + 2430^') x'^ 



+ (3807/?+ 6480?r) or' ^+ (393j9*+ 5076pV+ 2673^5* + 4320r') x" 



+ (594p'5 + 7I28i??r + 729?0:r'' !-(74) 



+ (33/ + 792/r + 243/9^ + 4752pr' + 29I69V) x" 



+ (27pV + 648pV + 3888?r') x' 



+ / + 36/r + 432pV' + I728r'} ; 



and, finally, 



(/.' - \=')'^ = (^^ - e' + 3et)^ _ A? = 



— 2-"^ 3-' { 125x'" + 350py "• + 400yy» + (285/ + 450r) x'^ 

 . + SSOp^-a^" + (32p» + 790pr + 410y') y" + (4 1 4/y + 9609r) ^ 



+ (— 1 6/ + 192/r + 546py' + 565r'') or'* 



+ (— 8/9 + 966pyr + 108«7^) ^" ^ ^^^^ 



+ (12/— 132/r + 105py + 464pr" + 522yV) x"" 



+ (S/gr — 48pV + 54p?^ + 576yr') x' 



+ l6pV — 4py _ 128/r=' + 144pyV + 256/^— 27y'}. 



23. This last result may be verified, or rather proved anew, and at the same 

 time put under another form, which we shall find to be useful, by a process such 



