MR. A. R. FORSYTH ON A CLASS OP FUNCTIONAL INVARIANTS. 
77 
will, for all values of m and n, be an absolute invariant. Thus, taking in succession 
m = 1 and = 0, and m = 0 and n = 1, we have 
-^ 10'^11 ^ 01^20 
and 
^10^02 
"10 
0- 
10 
the former of which, multiplied by ZqiAio subtracted from the latter, gives 
^%hohi ~t ho^^o2 
no 
as an absolute invariant, or ^10^ ^20 2^01 ^10 ^11 as a relative invariant, for the 
present transformation. 
General Transformation. 
3 . We now proceed to the consideration of functions which are invariantive for the 
general simultaneous homographic transformation of the independent variables repre¬ 
sented by 
_ y _ 1 _ 
aj + / 3 iX + 7 jY aj + /SaX + YjY «3 -|- ySgX + 73Y 
As it will be convenient to have some one invariant at least, a relative invariant for 
these transformations can be obtained as follows. An integral relation given by 
a hx + cy _ u 
a' + h'x -f- c'y v 
reproduces itself in form when the independent variables are subjected to the above 
transformation ; and the differential equation which is the equivalent of this integral 
relation will, therefore, also reproduce itself, and so will furnish an invariant. 
Now, both u and v satisfy the three equations 
— - n — 0 — - 0 • 
dx^ ’ dxdy ’ 87^ ’ 
and therefore, substituting vz as the value of n in these, we have 
0 = z^qV + 22:10^10, 
0 = Z^^V + 2io% + Zoi'*^10> 
0 = + 22opyoi- 
The elimination of v, v-^q, between these leads to the result 
