 First subsidiary maximum of a large slit array

For $N$ equal narrow slits, adjacent phase increment $\delta$ gives amplitude $\sin (N\delta /2)/\sin (\delta /2)$ relative to one slit. Near a principal maximum, set $u=N\delta /2$; for large $N$ the normalized intensity approaches $(\sin u/u{)}^2$. Approximating the first subsidiary maximum by $u=3\pi /2$ gives principal-to-subsidiary intensity ratio $9{\pi }^2/4$. The exact large-array extremum satisfies $\tan u=u$ in $(\pi ,2\pi )$ and gives a slightly different ratio, about $21.2$. The slit envelope can further modify observed peak heights.