SchlomilcJis Theorem in BesseVs Function*. 567 



spectrum — then for the orange-yellow the resulting light 

 should be compounded of these two; and the resulting light 

 should be plane-polarized, not depolarized. 



The depolarization is in fact only apparent; for on using 

 a moderately high power objective it is at once evident that 

 there is a structure in the wing-case which causes a difference 

 of phase between the components varying very rapidly from 

 point to point ; and the resulting plane of the plane-polarized 

 light varies with corresponding rapidity, leaving no trace of 

 polarization when the observation is made with a telescope. 



The absorption coefficient for this specimen is quite of the 

 order of that of the metals; and the thickness of the 

 " metallic " film is of the order of a ten thousandth of a 

 millimetre. 



I take this opportunity of expressing my appreciation of 

 the courtesy of Messrs. R. M. Strong, V. E. Shelford, W. L. 

 W. Field, and H. B. Ward, to which I am indebted for 

 bringing the literature of the subject to my notice, and for 

 the specimens on which these observations are based. 



Ryerson Laboratory, 

 University of Chicago. 



LXII. On a Physical Interpretation of S chid mil cli s Theorem 

 in BesseVs Functions. By Lord Rayleigh, O.M., F.B.S* 



THIS theorem teaches that any function f (r) which is 

 finite and continuous for real values of r between the 

 limits r = and r = 7r, both inclusive, may be expanded in the 

 form 



f(r)=a + a l J (r) + a 2 J (2r) + a li J (3r)+ . . ., . . (1) 



J being the Bessel's function usually so denoted; and 

 Schlomilch's demonstration has been reproduced with slight 

 variations in several text-books f . So far as I have observed, 

 it has been treated as a purely analytical development. From 

 this point of view it presents rather an accidental appear- 

 ance ; and I have thought that a physical interpretation, 

 which is not without interest in itself, may help to elucidate 

 its origin and meaning. 



The application that I have in mind is to the theory of 



* Communicated by the Author. 



t See, for example, Gray & Mathews 1 ' Bessel's Functions,' p. 30 ; 

 Whittaker's ' Modern Analysis,' §165. 



