The corresponding values of C and S were originally derived by A. L. Cauchy, without the use of Gilbert’s integrals, by direct integration by parts.
From the series for G and H just obtained it is easy to verify that
| dH | = − πvG, | dG | = πvH − 1 (18). |
| dv | dv |
We now proceed to consider more particularly the distribution of light upon a screen PBQ near the shadow of a straight edge A. At a point P within the geometrical shadow of the obstacle, the half of the wave to the right of C (fig. 18), the nearest point on the wave-front, is wholly intercepted, and on the left the integration is to be taken from s = CA to s = ∞. If V be the value of v corresponding to CA, viz.
| V = √{ | 2(a + b) | } · CA, (19), |
| abλ |
we may write
| I² = ( ∫ | ∞ | cos ½πv² · dv ) | ² | + ( ∫ | ∞ | sin ½πv² · dv ) | ² | (20), |
| v | v |
or, according to our previous notation,
I²=(½ − Cv)² + (½ − Sv)² = G² + H² (21).
| Fig. 18. |