This exercise was given in the first pages of Synge & Schild Tensor Calculus.
The parametric equations of a hypersurface in $V_N$ are
$x^1=a\cos{u}$,
$x^2 = a\sin{u^1}\cos{u^2}$,
$x^3 = a\sin{u^1}\sin{u^2}\cos{u^3}$,
$\cdot\hspace{0.2cm}\cdot\hspace{0.2cm}\cdot\hspace{0.2cm}\cdot\hspace{0.2cm}\cdot\hspace{0.2cm}\cdot$
$x^{N-1}= a\sin{u^1}\sin{u^2}\sin{u^3}\dots\sin{u^{N-2}}\cos{u^{N-1}}$,
$x^N=a\sin{u^1}\sin{u^2}\sin{u^3}\dots\sin{u^{N-2}}\sin{u^{N-1}}$,
where $a$ is a constant. Find the single equation of the hypersurface in the form $$F(x^1,x^2,x^3,\dots,x^N)=0.$$
The number of parameters is one less than the number of equations, so how to eliminate them to give the equation in the form above?
Sorry if this is too easy. I just can't see it.