12 Proceedings of the Royal Irish Academy. 
function of the single variable C Hence, when the single equation 
^ = 0 is fulfilled, we must have 
dU ,dV ^ dU ,dV ^ dU .dV ^ 
dx dx dy dy d% d% 
Hence, again, if ^ be a double root of the equation A = 0, these three 
equations can be satisfied by one of the variables, the other two re- 
maining arbitrary. If the general expression for 77 be 
ax^ + hy"^ + + 2ly% + 2m%x + ^nxy^ 
we have, then, the system of equations obtained by equating to zero 
€very single first minor of the determinant A, or 
a - \ n m 
n b - X I , 
m I c - X 
where X is a double root of the equation A = 0. 
At first sight it would seem that this method of investigation might 
be extended ; but if we assumed in general, as is assumed in the par- 
ticular case above, that two quantics of the second degree could 
always be transformed, one into a sum of squares of the variables, the 
other into a sum of squares of the same variables multiplied by con- 
stant coefficients, we could show, in the same manner as above, that 
any two conies which touch must have double contact. For, if they 
touch, the equation got by equating to zero the discriminant of XI -XV 
( 27"= 0, and F= 0, being the equations of the conies) must have equal 
roots, in which case, as shown above, if A be the value of the double 
root, U - AV must be the square of a linear function of the variables ; 
whence the conies must have double contact at the two extremities of 
the line represented by equating this linear function to zero. The 
absurdity of this result shows that we have no right to assume without 
proof the possibility of the transformation in question. 
Enough has been said to lead us to conclude that the problem before 
us possesses a considerable amount both of interest and difficulty. 
The harmonic determinant in its most general form is obviously the 
discriminant of U-XV^ where 27" and Fare each quantics of the second 
degree in n variables. 
In the present case, moreover, when we substitute in Ffor the 
