he accompanying table.

+--------------------+----------+------------+
| | V | I^2 |
+--------------------+----------+------------+
| 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,

_ v _ v
/ /
x = | cos 1/2[pi]v^2.dv, y = | sin 1/2[pi]v^2.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 = [+-][oo].

The intrinsic equation, expressing the relation between the arc
[sigma] (measured from O) and the inclination [phi] of the tangent at
any points to the axis of x, assumes a very simple form. For

dx = cos 1/2[pi]v^2.dv, dy = sin 1/2[pi]v^2.dv;

so that
_
/
[sigma] = | [sqrt] (dx^2 + dy^2) = v, (30),
_/

[phi] = tan^-1 (dy/dx) = 1/2[pi]v^2 (31).

Accordingly,

[phi] = 1/2[pi][sigma]^2 (32);

and for the curvature,

d[phi]/d[sigma] = [pi][sigma] (33).

Cornu remarks that this equation suffices to determine the general
character of the curve. For the osculating circle at any point
includes the whole of the curve which lies beyond; and the s