234 
MR. AIRY ON THE THEORETICAL EXPLANATION OF 
The uniformity in the progress of the values, both of G ( s ) and of <p ( s ), is very re- 
markable ; and seems to render it probable that, although the functions D ($) and 
E (s) have been formed in such different ways from C ( 5 ) and S (s), and although 
C (s) and S ( 5 ) are formed in a manner which scarcely permits a simple relation be- 
tween them, yet some simple expression for G (s) and <p (s) must exist. This pre- 
sumption becomes stronger, when we examine the values of s which correspond to the 
quadrantal values of <p ( 5 ). If we make <p (s) successively equal to 0, 90°, 180°, 270°, 
&c., and take the corresponding values of s, and the squares of those values, we form 
the following series. 
Quadrant of <f> (s'). 
Value of s. 
Value of s 3 . 
Quadrant of 0 (*). 
Value of s. 
Value of s 2 . 
0 
0-00 
0-00 
16 
3-94 
15*5 
I 
0-73 
0-53 
17 
4-06 
16-5 
2 
1-27 
1*61 
18 
4-18 
17-5 
3 
1-63 
2-56 
19 
4-31 
18'6 
4 
1-88 
3*53 
20 
4-42 
19-6 
5 
2-12 
4-50 
21 
4-53 
20-6 
6 
2-36 
5-60 
22 
4-64 
21-5 
7 
2-57 
6-60 
23 
4*75 
22-5 
8 
2-74 
7-50 
24 
4-85 
23-5 
9 
2-91 
8-49 
25 
4-95 
24-5 
10 
3-09 
9-5 
26 
5-06 
25-5 
11 
3-25 
10-6 
27 
5-15 
26-6 
12 
3-40 
11-6 
28 
5-24 
27-5 
13 
3-53 
12-5 
29 
5-34 
28-5 
14 
3-67 
13-5 
30 
5-44 
29-6 
15 
3-81 
14-5 
The values of s 2 corresponding to the successive quadrants of <p (s) increase pretty 
uniformly every time by 1, except in the first instance, where the increase is 0 - 5. This 
interruption of the law is not without analogues in Physical Optics. I may mention 
the expression for the intensity of light produced by a grating before a lens, namely 
( " s i n lO J w ^ich the first and greatest maximum occurs when 0=0, and the suc- 
ceeding maxima occur nearly v/hen 0 
71 O It 5 7T c 
— ? -X 5 5 &C. 
2 n 2 n 2 n 
The small irregularities in the progress of these various numbers may probably 
arise, as I think, from irregularities in Fresnel’s calculation of the original integrals. 
Several years ago I verified a portion of Fresnel’s table ; and though the agreement 
of my numbers with Fresnel’s was sufficiently close to show that the numbers might 
be used with perfect safety, yet there were some small discordances which seemed to 
indicate that a complete recalculation might be useful. 
Adopting now for 1 — 2 (C (s)) 2 — 2 (S (s)) 2 and 2 C (s) — 2 S (s), the expres- 
sions G (s) cos <p (s) and G (s) sin <p (s), and observing that 1 -f 2 (C (s)) 2 -f- 2 (S (s)) 2 
— 2 — G (s) cos <p (s), the expression for the intensity of light, on the point of the re- 
