ME. A. CAYLEY ON TSCHIENHAUSEN’S TEANSFOEMATION. 
563 
Similarly, if the second operator is represented by 
V — (2CdD-|"Bdc), 
then we have 
(a, 1), c, djw, c, d, ejx, 1)^=0, 
which by means of the equation 
[a, h, c, d, e^jc, 1)^=0 
is reduced to Vx=x'^. Hence in VV the part depending on the variation of x is 
ax'^'Q + + ihx‘^)C + ( ?)ax^ + %hx^ -\-6cx'^)D, 
and the other part of W is at once found to be 
and, adding, the coefficient of D vanishes on account of the quartic equation, and we 
have 
V V= {ax’^ + ihx + 3(?)B + 2[ax ^ + + 6 + 3 6?)C, 
which is precisely equal to 
(2CB„+BBe)V, 
so that V is reduced to zero by the operator 
V-(2CS:,+BBc), 
which completes the demonstration ; and the demonstration in the general case is pre- 
cisely similar. 
In the case of the cubic equation we have 
{a, b, c, d'Jx, 1)®=0, 
y={ax-\-h)B-\-{ax‘^-\-ohx-\-2c)Q ; 
and writing the second equation in the form 
(y_jB-2cC)+^( _^^B-35C)4-^’^( -aC) = 0, 
multiplying by x and reducing by the cubic equation, we have 
dO -\-x{y—l)Q-{- cC)-{-x% — aB)=0, 
and repeating the process, 
dB -[-^( 3cB+ dC)-\-x~{y-\-2hB-\-cC)~^ 
or, what is the same thing, we have the system of equations 
( y-hB-2cQ, - oB-UC, -aC Jl, .t, .r^)=0, 
dQ, y— hB-\- cC, —a B 
dB , 3cB+ dC, y+^hB^cQ 
and the resulting equation in y is of course that formed by equating to zero the deter- 
minant formed out of the matrix in this equation. The developed expression is 
( 1 , 0 , €,Wly. 1 )^= 0 , 
