566 
ME. A. CAYLEY ON TSCHIENHAUSEN’S TEANSEOEIklATION. 
I wi’ite 
U'=«B^^-4JBC^-c(2BD+4C^)+4^^CD+eD^ 
B!={ac-h’^)W-\-^ad-hc)^C+{ae-2hd+(f)B'D-^4:{hd-(f)0 
-\-2{he—Gd)CT)-\-{ce---d‘^)iy; 
and I represent by O' the expression which has just been found for These 
functions, U', H', O', are invariants of the two forms 
{a,h,c,d,eJK,Y)\ (B, C, D^Y, -X)^ ; 
we have, moreover, the invariants 
ae—ihd-\-?)C^, ace—ad^—h^e-\-2hcd—c^^ 
which I represent as usual by I, J, and the invariant BD—C^ which I represent by 0'. 
This being so, we have 
C=6H'-2I0', 
B=40', 
e=IU'='-3H'Hl^0'"+12J0'U'+2I0'H', 
the last of which may be verified as follows: — ^viz. writing «=e=l, J=(Z=0, c=^, it 
bGComGs 
(1 + 3 ^^){BH^( 2 BD+ 4 C^)+D^}" 
-3{^B"+(l+^")BD-4^"e+^D"}" 
-1-(1 + 3^^XBD~C=')^ 
+12(^-^^}(BD-e){^BH(l + ^*)BD-4W+^D^} 
= B^ 
H-B^C (12^) 
4 -B^C= ( - 6 ^+ 54 ^^) 
( 2 +36^^) 
+BeD( - 4 +36^^) 
+B^ (1-18^^+81^^) 
+ ( - 6 ^+ 54 ^^) 
+0D (12^) 
+D^ 
which is an identical equation. 
The expression for the invariant I (quadrinvariant) of the function (1, 0, C, 23, 1)^ 
is 0+3(■^C)^ or (0+3(H'— ^I0')^ viz. it is 
IU'^-3H'^+ P0'^+12J0'U'+2I0'H' 
+ 3H'^+ir0'^ -2I0'H', 
01’, finally, it is 
IU'2+|P0'2+i2J0'U', 
