ME. A. CAYLEY ON THE ANALYTICAL THEOEY OF THE CONIC. 
643 
at pleasure without altering the value of the expression. For instance, assuming for 
(x', y', z') the values (1, 0. 0), (0, 1, 0), (0, 0, 1) successively, the different values of the 
expression are 
ax + hy+ gz hx + by +fz gx +fy -1- cz 
V a Vl) V c 
But if, as assumed, {a, ..yx,y, zf be a perfect square =[ax-\-^y-^yzf, then 
{a, b, c,f, g, A)=(a^ f3\ /, (3y, ycc, a^), 
and each of the foregoing values becomes equal to the root ax-\-^y-\-yz. It is some- 
what singular that it is not possible to obtain symmetrical formulse without employing 
in this manner supernumerary arbitrary quantities such as [x', y\ z'). 
14. Next, if (a, ...'Xx,y,zy, instead of being a perfect square, only breaks up into 
factors, then in the foregoing identical equation the right-hand side is a perfect square, 
and by the formula just obtained its value is 
[(A, ..JX, Y, Z'^yz'—y’z, zx'—z'x, xy'—x'yyy 
(A, ..IX, Y, Z)^ 
where (X, Y, Z) are supernumerary arbitrary quantities. The identical equation then 
gives 
{a, ..I^, y, zf — 
y, "A*’ 3/. + (A,..lx,x,z)= 
and consequently 
(a, ..Jx, y, Product of 
(a, ..I^, y, zjx', y, z'}± (^’ ZX y^]/0^^x'-z'x, xy'-x'y )\ 
^-(A,..IX,Y,Z)^ 1 
a formula which exhibits the decomposition of (a , . y, zf assumed to be a function 
which breaks up into factors; the formula contains the two sets of supernumerary 
arbitrary quantities {x’,y\ z') and (X, Y, Z). It will be remembered that (A, . .) denotes 
the system of inverse or reciprocal coefficients {bc—f '^, . ..). 
16. Consider the formula 
(«, b, c,f, g, ^1'— I;?'— !■';?/= (a, b, c, f, g, hll', 
which gives 
a= 
b= -2r/^|, 
c= b^^ -farj^ 
t = +#?,+/, 55 , 
g=—Ki+fh —gif’ 
*1'. C)^ 
