116 
If now the cubic, f, = 0, has a double point, then the first polar: 
Df, =o0F 
dfs d fs d fs 
that is: ae ia Sao 
where /, has been made homogeneous by the introduction of the variable z. 
If 7, =o has a double point then the curve is a unicursal or rational curve. 
Indeed, if one solution of X (x, 7) =o is rational, the other must also be rational, 
and hence the cubic, /, = 0, is unicursal, and hence it must have a double point, 
since its deficiency is zero. 
Theorem: The integral : 
{ F (x, y) dx 
eet yc 
can be reduced to a rational integral only when the cubic : 
Sf; =y? — R, (x) =0 
is a wnicursal curve. 
Consider next the general integral: 
{ F (x, y) dx 
‘ m a 
y=V Rn (x), m =n, 
The curve: 
jan =9y™ — Ra (2) =0 
is of the n™ order, and the equation: 
K (z, A)=0 
is of the (n—1)** degree. If one rational solution of this equation can be found 
the reduction can be made, otherwise not. 
Suppose the curve: fn —o0, has a multiple point of order k, then, by taking 
A at this point, the equation 
X (a, 2) —0 
is of degree (n-k). In this case it is necessary to find a rational solution of an 
equation of the (n-k)™ degree. Now fn =o, has a multiple point of orner k if 
the (k-1)** polar of that point vanishes identically. 
pe fn = 0: 
If f, =o has a multiple point of order (n-1), that is if: 
DD” fn —— is) 
then the equation : 
X (x, 7) =o0, 
is linear and the reduction is always possible. 
*“ Clebsch. Vorlesungen iiber Geometrie. Vol.I, p.315. 
