Composition Theorem

Let

$\begin{displaymath} Q(x,y)\equiv x^2+y^2. \end{displaymath}$

Then

$\begin{displaymath} Q(x,y)Q(x',y') = Q(xx'-yy',x'y+xy'), \end{displaymath}$

since

$\begin{eqnarray*} (x^2+y^2)(x'^2+y'^2)&=&(xx'-yy')^2+(xy'+x'y)^2\\ &=&x^2x'^2+y^2y'^2+x'^2y^2+x^2y'^2. \end{eqnarray*}$

See also Genus Theorem