The second and third factors of (3) being each of the form sin²u/u², we have to examine the character of this function. It vanishes when u = mπ, m being any whole number other than zero. When u = 0, it takes the value unity. The maxima occur when
u = tan u, (4),
and then
sin²u / u² = cos²u (5).
To calculate the roots of (5) we may assume
u = (m + ½)π − y = U − y,
where y is a positive quantity which is small when u is large. Substituting this, we find cot y = U − y, whence
| y = | 1 | (1 + | y | + | y- | + ...) − | y³ | − | 2y5 | − | 17y7 | . |
| U | U | U² | 3 | 15 | 315 |
This equation is to be solved by successive approximation. It will readily be found that