MR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 
647 
T =1—20/'— 8/®. 
R =_ (1-1-8/')'. 
HU =/V+3/'+2')-(1+2/')^^2. 
0U = ( 1 +8/')'(3/'2'+2'a:'-|-,r'3/') 
+ ( — 9/®) (^'+3/' + 2 ')' 
+ ( — 2/ — 5/^— 20/^) {x^-\-y^-\-z^)xyz 
4- ( — 1 5/' — 78/' + 1 2/')a?y z'. 
0,U =4(l + 8/')'(3/'«'+a'.*'+<r'3/') 
+ (- l-4/'-4/®)y+y+2')' 
+ (4/+ 100/'+ 1 12/7)(^'+3/'+«')xyz 
+ (48/'+552/'+48/').i?'3/V. 
0^,U= — 2(1 +8/')'(3/';z'+«'a;'+a;'3/') 
+ ( 1 — 1 0/') ( j?' +?/' +2')' 
+ (6/ — 180/' — 96/^)(j:'+3/'-|- z^)xyz 
+ (6/'— 624/'— 192/').^■'^/V. 
PU + 
QU = ( 1 - 1 0/') (1'+^'+^') - 6/'(5 + 4/')!;?^. 
FU =-4(l+8/')(;j'^'+^'r+f;j') 
+ (f+^^ + ^^)^ 
-24/'(f+^'+^')|;,^ 
-24/(l+2/')|V^', 
to which it is proper to join the following transformed expressions for 0U, 0;U, 0^,U, 
viz. 
0U =(1 +8/')'(3/'2'+2'^'+:j;'3/') 
+ (-2/'-/®)U' 
+ (2/-5/')U.HU 
+ (-3/' )(HU)'. 
0^U =4(1 +8/')'(3/'2'+2:'^'+a?'3/') 
+ (— l + 12/'+4/®)U' 
+ (-16/+4/' )U.HU 
+ (-12/' )(HU)'. 
0,^U= — 2(1 +8/')'(3/'«'+2'j?'+.'r'3/') 
+ (l_i6/'-6/®)U' 
+ (6/ )U.HU 
+ (6/' )(HU)'. 
The last preceding table affords a complete solution of the problem to reduce a 
ternary cubic to its canonical form. 
