N 



1^-2-^2 



— ^wi2 ^m 



i=di m =(S 







h — h 



=di =« + fi 





di m 

 di 



=/'(0 



*+/3 







/(O 



C< /3di 

 Jo « + # ' ' ' 



cm<f Dispersion in certain Metals. 369 



lelism of both pencils cannot in any way be interfered with by 

 anything happening at the bounding surfaces a, 6, c. In fact 

 this parallelism is only destroyed by refraction at the fourth 

 boundary metal | air ; therefore our experiments can only 

 teach us something about what happens at this boundary, in 

 particular about how i depends upon i m . For our purpose it 

 is, however, better to consider ? the independent variable, as 

 this angle is the one directly measured. 



We therefore put i m =f(i), evidently a single-valued odd 

 function, unknown as yet. From the diagram in fig. 1 it 

 follows at once that 



and 



and /(0 =) o ^. . . . (1) 



§ 9. Now from our measurements we know the values of a, 

 and therefore also of /3/(a + /3), for a set of values of i. We 

 therefore in a certain sense obtain an experimental differential 

 equation of the simplest kind, which we have only to integrate 

 in order to obtain the relation sought for between i m and i. 

 This we have actually carried out ; with abscissas i values of 

 /3/(a + /3) were plotted as ordinates, the points thus obtained 

 joined by a smooth curve, and the values of the integral / (i) 

 arrived at by graphical integration, starting from the initial 

 value /(0) =0, evident for reasons of symmetry. 



§ 10. If the prisms be for a moment supposed to consist of 

 ordinary transparent matter of index ?i, Snellius's law would 

 hold, i. e. 



i„=/(i)=sm-'(Sp), .... (2) 

 and this would give by differentiation the explicit equations 



and 



«=/3<f^ZE5Zl-l"l (4) 



L cos i ) 



which will bo repeatedly used below for purposes of calcula- 



