| − | 2lh | · | ƒ | sin | kηl | · | 2ƒ | sin | kξh | · sin k{at − ƒ − R + | ξh | } (2) |
| λƒ | kηl | ƒ | kξh | 2ƒ | 2ƒ |
If we put for shortness π for the quantity under the last circular function in (1), the expressions (1), (2) may be put under the forms u sin τ, v sin (τ − α) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin τ and cos τ in the expression
u sin τ + v sin (τ − α),
so that
I = u² + v² + 2uv cos α,
which becomes on putting for u, v, and α their values, and putting
| { | ƒ | sin | kηl | } | ² | = Q (3), |
| kηl | ƒ |
| I = Q · | 4l² | sin² | πξh | {2 + 2 cos( | 2πR | − | 2πξh | )} (4). |
| π²ξ² | λƒ | λ | λƒ |
If the subject of examination be a luminous line parallel to η, we shall obtain what we require by integrating (4) with respect to η from −∞ to +∞. The constant multiplier is of no especial interest so that we may take as applicable to the image of a line
| I = | 2 | sin² | πξh | {1 + cos( | 2πR | − | 2πξh | )} (5). |
| ξ² | λƒ | λ | λƒ |