98 
PROFESSOR CAYLEY ON THE CURVES 
(2, 1, 1, l) = 6m 3 +30m 2 w+30wm 2 +6w 3 — 174m 2 — 348mw— 174?t 2 +1320m+1320?t 
+a(^m 3 4-m 2 w+mw 2 +^ 3 — i 2 1 w 2 —26mw— J ^i 2 960) 
+a 2 ( — fm— -fw+28); 
(1, 1, 1, 1, 1)*= m»i 5 + tS- W + £m V -1- -^mn* + xio^ 5 
— — %m 3 n — 2mV — %mn 3 — Y^n 4 
— 1 Yrm 3 — ~Yi~m 2 n — -^mn 3 — -^n 3 
+ J^| 7 m 2 + 59 ^ + i ^ % 2 _ 3 I^_ 3 J^ 
+ a( — im 3 — f m 2 w — -fm% 2 — ^t 3 +\-m 2 + 2 3 m +-^w 2 — ■ a f 2 w — + 486) 
+ a 2 (fm+frc— 15); 
51. I observe that by means of the abovementioned expressions of (X, 4Z) and 
(2X, 3Z), the foregoing results, other than those for (5), (4, 1), &c., may be presented 
in a somewhat different form, viz. we have 
(4Z)(1)=< ■ )+m(/), 
where ( • ) denotes (4Z •),(/) denotes (4Z /), and so in other cases, the understood term 
being 3Z or 2Z, as the case may be. 
(3Z)(2) = (•/»«; 
(3Z)(1, 1) = 
+ (- / )(mn a) 
+( //)(fyn 3 — \m ) ; 
(2Z)(3) = ( .'. )( fm + n- fa) 
+ ( : / )( — 'fan— f^+fa) 
+ ( ■ / /){~^ ,n ~'k ri "hf 04 ) 
+(///)( m+>— |a); 
(2Z)(2,1) = (.*. ){-3m-3 w + a (_>+^+2)} 
+ ( : /){ fm+fw + a( fm+>— 4)} 
+ (•//){ |w+|^+a( |m+fw — 4)} 
+ ( ///){ — 3m— 3w+a( £m-> + 2)} ; 
(2Z)(1, 1,1) = 
( .-. ){— -^m 3 — |m 2 }i+|m» 2 4-^ 3 + -g-m 2 + 0m?t— f ^ 2 +ri^+fl^+ a ( fw— f 1)1 
+ ( :/){ r^ 3 +l^+imw 2 — j^ 3 — j^m 2 — iww+i^ 2 — ¥^+<— 
+ ( • //){ iV^ 3 +i^ 2 ^+|w^ 2 +i L 6^ 3 +i% m2 -‘i w ^— -x w ~ J ^+ a (“T^ m "“'5V 4 + 3 )} 
+ ( ///){ ^m 3 +-jm 2 w— |m» 2 - ^* 2 — -fm 2 + 0??m+ ^ w 2 +f-fm+y^+a(— f m+ fw— 1); 
in all which formulae it is to be recollected that we have 
(■*■)-« ;/)+!( 7/)-(///)=o. 
* In my paper in tlie Comptes Rendus, I gave erroneously the coefficients — - a s 59 m— 32 °° n.. +a(... + ++. 
