28 ELEMENTARY PROPERTIES OF CURVES Booh I 
We next ask whether this same curve could be written in 
more than one way. This is the case. We write 
#'+#' = #'+#' 
cf)' = 1/1' = ip'—dip. 
Here 6 is a general polynomial of degree n—{n x -\-n 2 ). The 
reasoning is reversible, the real freedom of a curve compounded 
out of cf) and ifj in this way is the apparent freedom, less the 
freedom of 6. The real freedom of the compounded curve will be 
{n—n 2 +l){n—n 2 +2) ^^(^—1) {n—n 1 +l){n—n 1 +2) 
2 Z, 2 ^ 2 
i 
V)_{n— {n x +n 2 )-\-\][n—{n 1 +n 2 )+2] _ 
Z, 2 ~ " 2 
2 2 
(w+l)(w+2) V r *fo— 1) !) 
-n x n 2 —1. 
But 7l x Yl 2 = 2 Vi, 
i 
since either gives the total number of intersections of $ and i/;. 
Hence the amount of freedom is 
{n+l){n+2) y 1) _-, 
2 Z/ 2 
and this, as we saw above, is exactly the freedom of /. We have 
thus shown that if the order of / be sufficiently high, it can be 
compounded out of «¡6 and i/j in exactly this way. 
What will happen when the order is less high ? It is con 
ceivable that there are some curves of order greater than n x -\-n 2 
which fulfil the conditions at the intersections of <f> and ifj but 
do not take this form. Let / be such a curve of the highest 
possible order where this compound form is not obligatory. It 
will be obligatory if we multiply / by a linear expression, i.e. if 
ax-\-by-\- c = 0 
be a straight line not through any intersection of </» and i/j 
{ax-\-hy-\-c)J= #'+#'