( 



Anomalous I^ispersion and- Absorption of Fuchsin. 213 



hero are not substantially new, it appeared of some interest 

 to show how they could be deduced, in a simple manner, 

 from the general solution of Laplace's equation. The method 

 seems to me to thron' some new light upon the relation of 

 the Legendre to the Bessel functions. The latter, according 

 to the view here set forth, should be looked upon as in some 

 ways the more fundamental of the two. 



There are two papers which deal with the expression of 

 Legendre functions in terms of Bessel's functions. One of 

 these is by Mr. Schafheitlin (Math. Annalen, vol. xxx.) on 

 the representation of a hypergeometric function by definite 

 integrals. Mr. Schafheitlin obtains the hypergeometric func- 

 tion as a detinite inteoral involvino- a BessePs function under 

 the integral sign. The Legendre function is then obtained 

 as a special case of the hypergeometric function. 



The other paper is by Mr. Steinthal " On the Solution of 



the Equation {l — x'-) -^^ — 2a^ — +n[n-\-l)u = '^ (Quarterly 



Journal of Pure and Applied Mathematics, vol. xviii.). The 

 Legendre coefficients Pn and Q„ are obtained as definite 

 integrals involving the Bessel's function of order n-\-^. 



Neither Mr. Schafheitlin nor Mr. Steinthal makes any use 

 of what I liaA'e called "• Bessel operators/' and neither has 

 considered the associated Legendre's coefficients. 



I understand from Mr. G. F. C. Searle. M.A., of St. Peter's 



College, Cambridge, that the expression J J ^i_^2 j^« 



for P«(/a) has been previously arrived at by Mr. Oliver 

 Heaviside, by considering the series for Pn(yLfc) directly. This 

 result, however, is now seen to be merely a particular case of 

 a more general theorem. 



XX V^. The Anomalous Dispersion and Selective Absorption of 

 Fuchsin. By W. B. Cartmel, M.A., former Fellow in 

 Physics at the University of S el rrasha^ . 



[Plates IX. & X.] 



ALTHOUGH fuchsin is the substance in which anomalous 

 dispersion was first observed, and although it shows 

 this phenomenon much more decidedly than any other sub- 

 titance upon which it has been found possible thus far to make 

 anything like reliable measurements, its optical constants 



* Read before the Washington meeting of the American Associa- 

 tion for the Advancement of Science. Communicated by Prof. D. B. 

 Brace. 



