564 
]\IE. A. CAYLEY ON TSCHIENHAIJSEN’S TEANSEOE^^IATION. 
where 
BC 
Q? 
ac 
+ 1 
ad 
+ 1 
hd 
+ 1 
¥ 
— 1 
be 
— 1 
— 1 
+ 1 
+ 1 
+ 1 
* 
B* 
B^C 
BC^ 
C® 
drd 
+ 1 
ahd 
+ 3 
acd 
— 3 
— 1 
abc 
— 3 
ac^ 
-6 
Pd 
+ 6 
bed 
+ 3 
P 
+ 2 
Pc 
+ 3 
bP 
-3 
—2 
4-3 
+ 6 
+ 6 
+ 3 
The sum of the coefficients in each column should here and elsewhere in the present 
memoir be equal to zero, and I have by way of verification annexed to each column the 
sums (+8' number) of the positive and negative coefficients. The coefficients C, 13, 
and therefore the function in y, are invariants of the two forms, 
(«, 5, c, Y)^ (B, CXY, -X); 
or in the present case, where there are only two coefficients B, C, the coefficients C, 23, 
and therefore also the function in y, are covariants of the single form («, 5, c, 
considering therein (B, C) as the facients. 
It may be remarked that in the present case, assuming the invariance of the function 
in y, we may obtain the transformed equation in a very simple manner by writing in the 
first instance C=0, this gives 
{a, 5, c, d\x, 1)^=0, 
y=[ax-\-h)^, 
and thence 
l(a, b, c, dXy—^B, «B)" = 0; 
or developing, 
y^ + dy{ ac — b‘^)W -\-{a^d—?>dbc-\-2 J®)B® = 0, 
wffiere the expressions for the coefficients are to be completed by the consideration that 
these coefficients are covariants of the form [a, b, c, 
case in hand of a cubic equation that the transformed equation can be obtained in this 
manner. 
In the case of a quartic equation, we have 
{a, b, c, d, ejx^ 1)^=0, 
yz=.[ax-\-b)Q-\-{ac^-\-ibx-\-?>c)C-\-{ax^-{-^bx‘^-\-^cx-\-^d)D, 
and these give the system of equations 
