80 
MR. A. R. FORSYTH OH A CLASS OF FLHCTIONAL INVARIANTS. 
Let z = 6 {x, y), and let the variables be transformed by the equations 
Let 
x=4> (X, Y), 2/ = V' (X, Y). 
^ ~ ^ (X p. Y “b O') — X, 
'F = r// (X + p, Y + O') — y, 
so that <I> and vanish with p and cr. Then, by the generalised form of Taylor’s 
Theorem, 
1 + 
ml n\ DX™ 0Y“ 
is the coefficient of p^o-" in the expansion in ascending powers of 
^ (X + p, Y + 0-), i/; (X + p, Y + 0-) 
d (x + d), y + ^), 
where p and cr occur only in $ and Now, 
and therefore 
1 + n' ^ 
^ ’ m' = o,r = o «b! n'l dy"' 
771 
I 0™ + ”Z 
! 77 ! 0X"* 0Y” 
][ 0m' + n'^ 
,,,?0 = 0 
where C^, n denotes the coefficient of p'^o-" in the expansion of in 
ascending powers of p and cr. When 77% and 71 both vanish, or when m -\-7i '>7ii-\-n, 
the coefficient C,„_ „ is zero. 
The form of the corresponding theorem for the case of any number of independent 
variables is evident. 
HoTrtiogi'a'phic Transfoi'mation: Chai^actei'istic Equations. 
7. When we consider the general homographic transformation, we may take 
ao to be zero, for the invariants do not explicitly contain x and y, but only differential 
coefficients with regard to them, and so they may be modified by the subtraction of 
the respective constants J then the general forms are equivalent to 
X _ y 1 
X + aY Y + ^X «3 + /S 3 X + TsY 
In order to apply the method of infinitesimal variation, it is sufficient to make the 
factor J nearly equal to unity, or, what is the same thing, to make x nearly equal to 
