From (24), (26) we see that the width of the bands is of the order √{bλ(a + b)/a}. From this we may infer the limitation upon the width of the source of light, in order that the bands may be properly formed. If ω be the apparent magnitude of the source seen from A, ωb should be much smaller than the above quantity, or
ω < √{λ(a + b)/ab} (27).
If a be very great in relation to b, the condition becomes
ω < √(λ / b) (28).
so that if b is to be moderately great (1 metre), the apparent magnitude of the sun must be greatly reduced before it can be used as a source. The values of V for the maxima and minima of intensity, and the magnitudes of the latter, were calculated by Fresnel. An extract from his results is given in the accompanying table.
| V | I² | |
| First maximum | 1.2172 | 2.7413 |
| First minimum | 1.8726 | 1.5570 |
| Second maximum | 2.3449 | 2.3990 |
| Second minimum | 2.7392 | 1.6867 |
| Third maximum. | 3.0820 | 2.3022 |
| Third minimum | 3.3913 | 1.7440 |
A very thorough investigation of this and other related questions, accompanied by fully worked-out tables of the functions concerned, will be found in a paper by E. Lommel (Abh. bayer. Akad. d. Wiss. II. CI., 15, Bd., iii. Abth., 1886).
When the functions C and S have once been calculated, the discussion of various diffraction problems is much facilitated by the idea, due to M. A. Cornu (Journ. de Phys., 1874, 3, p. 1; a similar suggestion was made independently by G. F. Fitzgerald), of exhibiting as a curve the relationship between C and S, considered as the rectangular co-ordinates (x, y) of a point. Such a curve is shown in fig. 19, where, according to the definition (5) of C, S,
| x = ∫ | v | cos ½πv²·dv, y = ∫ | v | sin ½πv²·dv (29). |
| 0 | 0 |
The origin of co-ordinates O corresponds to v = 0; and the asymptotic points J, J′, round which the curve revolves in an ever-closing spiral, correspond to v = ±∞.