(6),
We first find an approximate value for θ: then calculate P1, and find by (6) a more accurate value of θ; then, if necessary, recalculate P1, and thence θ, and so on.
II. Construction of Tables by Subdivision of Intervals
6. When the values of u have been tabulated for values of x proceeding by a difference h, it is often desirable to deduce a table in which the differences of x are h/n, where n is an integer.
If n is even it may be advisable to form an intermediate table in which the intervals are ½h. For this purpose we have
u1/2 = ½ (U0 + U1)
(7),
where
U = u − 1⁄8δ2u + 3⁄128δ4u − 5⁄1024δ6u + ...
= u − 1⁄8[δ2u − 3⁄16 {δ4u − 5⁄24 (δ6u − ...) } ]