260 PROFESSOR POWELL'S REMARKS ON THE 



which, on dividing by k'^, restoring the value of 6 (20.) and of k (2.), will give for the 



velocity (or-jj (on multiplying both numerator and denominator by (~) on the 



second side of the equation, and including all the coefficients in a new single term,) 

 the formula 



r sin'("-Af)-| 



?=H"'-^¥f' •'• • (^^-^ 



the same which has been deduced for unpolarized and plane polarized light. 



From this formula, on the supposition of a common coefficient, I made my approxi- 

 mate computations in Nos. I. and II. of my former Researches. On expanding the sine 

 and dividing by the arc there results a series of even powers of X, which, in a general- 

 ized form, is that from which Sir W. R. Hamilton deduced his elegant formula for 

 interpolation, by which I performed my calculations in No. III. From a similar series, 

 also, M. Cauchy has deduced the elaborate method of calculation which he has ap- 

 plied to Fraunhofer's indices in his Nouveaux Exercices de Math^matiques, and also 

 in his lithographed memoir on Interpolation: whilst the same series obtained by 

 Mr. Kelland, and modified by the introduction of the value of X in vacuo (that term 

 being in the above formula the wave-length in the medium), or 



■^=^+(iy<i-{ir^ — • • • • (3^-) 



is the formula which I employed and explained in No. IV. Precisely equivalent re- 

 sults are also deduced by Mr. Tovey. 



(11.) The term on which the dispersion depends, viz. 



sm I I . 



^'^ (35.) 



(^1' 



manifestly approaches to unity as A ^ diminishes in comparison with X, the applica- 

 tion of which was originally pointed out by Mr. Airy. The dispersion is insensible 

 in free space, and large in dense media. Hence Mr. Kelland has inferred that the 

 ether is in a state of greater density in free space, and less in dense media. We 

 shall recur to this point in the sequel. 



(12.) Experiment does not show that the state of a ray as to polarization, produces 

 any difference in the magnitude of its refractive index. Hence it follows that in the 

 formula (30.) the values of v^ must he constant for all values of b, as well as of u and&. 

 Hence for any particular value of h the several sums of terms involved must be sup- 

 posed to vary in magnitude, so that, joined with the condition (4.), the whole ex- 

 pression shall always remain constant, and equal to that in the case where b vanishes, 

 and where a and j8 are no longer restricted to the condition (4.). 



