A THEORY OF THE DISPERSION OP LIGHT. 251 



Here the value of ^ will evidently vary with a change in the value of X, or from one 

 rai/ to another ; it will also vary with a change in the constants H, r, or n, that is, 

 from one medium to another. 



The mere inspection of the formula will suffice to show that it exhibits at least a 

 general accordance with the obvious constitution of the prismatic spectrum in the 

 greater dispersion of the blue end. 



For, in general, as X is diminislied, the arc (-y-j is increased, and consequently 



the ratio of the arc to its sine increases, or (jb increases. And the variation in the 

 value of this ratio, and consequently in that of [ju, for a given variation in X, is greater 

 when the arc is greater, that is, when X is less. 



Thus, towards the blue end of the spectrum, where X is least, the dispersion or ex- 

 pansion of the rays is greatest. 



But we must proceed from these very general remarks to the more precise compa- 

 rison of numerical values. 



Comparison of Numerical Results. 



In proceeding to apply the formula to actual calculation, we are met by several 

 difficulties arising out of the peculiar form of the function. The process is, in fact, 

 reduced to finding arcs which shall fulfill the twofold condition of being themselves 

 in the ratio of the values of X, while they are to their sines in the ratio of the values 

 of (/J. For this I have not been able to make any direct method available. 



By indirect and tentative methods, however, and the assumption of arcs which 

 were seen (from a table of the lengths of arcs,) to be nearly in the required ratio to 

 their sines, I advanced by successive trials of greater or less arcs to more exact 

 values. Those for the two extreme rays were usually assumed in the first instance, 

 and their ratios to their sines compared with the ratios of the refractive indices; 

 and these once brought to a sufficiently near accordance, a fundamental arc was 

 obtained, from which those for the other rays were deduced on dividing by the 

 corresponding value of X\ and the product of a constant coefficient multiplying 

 the ratio of the arc and sine, which in theory ought to give the value of the refrac- 

 tive index, was compared with the index deduced from observation. This will 

 sufficiently explain the meaning of the several columns in the tabular statement of 

 the results. 



It must be borne in mind that the values finally adopted are still only approxima- 

 tive, and are open to further correction by repeating the process ; so that in all the 

 cases here considered a still closer coincidence might probably be obtained were it 

 thought desirable. 



The fundamental data of these comparisons are (as already said) those very precise 

 determinations of the value of X for the several definite rays named by the letters B, 

 C, D, &c., obtained by Fraunhofer from the interference-spectrum ; and which 



2 K 2 



