154 
Proceedings of the Royal Society of Edinburgh. [Sess. 
where J denotes the Jacobian of the four functions with respect to the 
four variables. 
A function 0, of any of the quantities subject to transformation, is said 
to be invariantive if, when precisely the same function <p' of all the 
quantities in (j> after they have been transformed is constructed, a relation 
is satisfied, jp being an integer. When p is zero, the function is said to be 
absolutely invariantive or (simply) an invariant ; when p is not zero, the 
function is said to be relatively invariantive. Generally, p is called the index. 
The simplest example of such a function connected with a quadratic 
differential form 
{a I dxy 
arises when we consider the discriminant A, where 
A = 
We have 
a = (^a, b, c, d, . . . 
}i = h, c, (/,... 
and so on ; and therefore 
A' = 
a , 
h. 
9, 
1 I 
h. 
b, 
m 
9. 
f, 
c , 
71 
h 
771 , 
71, 
d 
dx^ 
0^2 
dx.^ 
dx^' ’ 
dx^'" 
’ dx j ’ 
dx^ 
dx.2 
0:^4 y0.Tj 
dx^ ’ 
dx/ 
’ dx-^^xbx^ 
0^3 0a?4 
0 .^ 2 ' ’ 0 ^: 2 ' ’ dx^' 
n‘^(- 
^ JC.-\ 
\x 
1 , a.’2 , X3 , 
a, h\ g\ V 
h', b', f, ni 
g\ f, n 
r, m\ n, d! 
a, h, g, I 
h, b, f, m 
9, f, c, n 
I, 771, 71, d 
= AQ2. 
This function of the coefficients of the form is of persistent recurrence ; it 
will regularly be denoted by A. 
10. The infinitesimal transformations, the aggregate of which deter- 
mines the general transformations, are taken in the form 
