[yee a 
LIX. Theory of the Rainbow due to a Circular Source of Light. 
By K. Atcut, Rigakushi, and T. TANAKADATE, Rigakushi, 
Imperial University, Tokyo”. 
a IRY+ was the first to establish a satisfactory theory of 
the rainbow on the undulatory theory of light, based 
on the assumption of a point source of light. Airy, however, 
assumed that the equation of the meridian section of the 
emergent wave-surface referred to the tangent, and the 
ae aw? 
3a’ 
a being an undetermined constant. This point in Airy’s 
theory was afterwards developed in detail by Boitel t, 
Larmor §, and especially by Mascart || and L. Lorenz 4. 
Recently, the colours of the rainbow were minutely inves- 
tigated by Pernter **, by using Maxwell’s theory of compound 
colours. In this paper Pernter added a short calculation on 
the colours due to a source of finite dimension, by mere 
numerical addition of the results for the case of a point source; 
and he considered this to be sufficient to represent the colours 
due to the sun. Itis to be remarked that his values tf of 
Airy’s integral f#(z), on which the whole calculation is based, 
are sometimes discrepant from those originally given by Airy 
for <<4; and for z>4 we could not arrive at his results after 
repeated calculations. So far as we are aware, the various 
calculations are as yet limited only to cases which, strictly 
speaking, hold only for a point source of light. These con- 
siderations led us to undertake the following investigation. 
It may, therefore, be looked upon as an extension of Airy’s 
theory, when the apparent diameter of the sun is taken into 
account. 
Experiment must go hand in hand with theory. Miller tt 
and Pulfrich §§ verified Airy’s theory in the special case of 
two dimensions with a cylindrical stream of water (or glass 
rod), and a straight slit as the source of light. Buta question 
normal to the curve at the point of inflexion is y= 
* Communicated by Prof. Nagaoka. 
¥ Trans. Camb. Phil. Soe. vi. p. 379 (1838). 
t Compt. Rend. May 28, 1888; Phil. Mag. xxvi. p. 239 (1888). 
§ Proc. Camb. Phil. Soe. vi. p. 283 (1888). 
|| Zrawté d’ Optique, i. p. 382 (1889). 
*| Guvres Scientifiques, i. p. 405 (Copenhagen, 1898). 
** Wren. Sitz.-Ber. evi. 2 a, p. 135 (1897) ; Neues tiber den Regenbogen 
(Wien, 1898). 
+t Loe. cit., p. 140. 
tf Trans. Camb. Phil. Soc. vii. p. 277 (1841). 
§§ Wied. Ann. xxxiii. p. 194 (1888). 
