388 Lord Blythswood and Dr. Marchant on 



^ By differentiation with respect to X we have, after putting 

 sin 6 = 6 and cos 0=1, 



7nd\ = t(dfi/d\)d\ + sd6 (2) 



By differentiation with respect to m we have 



\dm = sd6 (3) 



Eliminating 5, and putting dm = \ so that 



h6 l = angular dispersion between two lines whose differ- 

 ence in wave-length is required ; 

 B6 2 = angular dispersion between two successive orders, 



we have 



B\(m-tdfi/d\)=\™r l (4) 



ou 2 



Substituting for m the approximate value (jx — l)t/X, 



(0»-i) 



d/j,\ 80,' 



X d\/ 



The value of X 2 t( (^— 1)— \-j£ \ can be calculated from 



the known refractive index of the glass ; and this constant 

 being known, the difference in wave-length between two lines 

 seen in the echelon may be determined, if their dispersion 

 relative to the distance between two successive orders be 

 known. It is to be noted that this constant is quite inde- 

 pendent of the width of the steps, and is therefore a constant 

 for the echelon, however it is put together. The only 

 necessary condition is that the instrument shall not he much 

 tilted, i. e. that the light shall pass through it in a direction 

 nearly normal to the plates. The values of this constant 

 were calculated for a series of different wave-lengths and a 

 curve plotted, as shown on fig. 3 (k ly 3). The expression 

 l/t((/x — 1)— XdfijdX) is much more nearly a constant for the 

 gratings, and for the observation of the Zeeman effect is 

 useful in giving at once the value of BX/X 2 corresponding to a 

 dispersion B6 lt 

 Thus 



BX 1 Sdt 



X 2 ' l , 1N ■ dfi\802 

 The curve for this expression is also shown in fig. 3 \k 2 , 4) 



