[ 116 ] 



XXVI. Remarks on the. Formula for the Dispersion of Light. 

 By the Rev. Badjcn Powkll, M.A., F.R.S , Savilian Pro- 

 fessor of Geometry, Oxford. 



[Continued from vol. viii. p. 309.J 



IN a former portion of these papers I have given some ac- 

 count of the methods by which the formula of dispersion, 

 or relation between the length of a wave and the velocity of its 

 propagation, or the refractive index for a given ray and given 

 medium, is made applicable in calculation. I have also illus 

 trated the comparison made by the researches which Sir Wm. 

 R. Hamilton has given me between tliat formula in its simpler, 

 but consequently only approximate, form, and the exact deve- 

 lopment. In the present instance, in continuation of the same 

 object of facilitating the study of this highly interesting por- 

 tion of the science of light, my design is, besides some general 

 remarks, to furnish certain constant elements which enter into 

 all calculations by the exact method ; together with one more 

 instance of the comparison of this method with the approxi- 

 mate one from the same source. 



The exact method, in fact, consists in this : The relation is 

 expressed by the series of powers of the reciprocal of A re- 

 sulting from the division of the development of the sine by the 

 arc, with certain indeterminate coefficients. In the approxi- 

 mate method these are supposed all constant and equal, and 

 are expressed by a common factor. In the exact development 

 this cannot be allowed; but an investigation is given by which 

 they are, in fact, eliminated ; and there results a method for 

 obtaining the theoretical index which is equivalent to the 

 enunciation of a law expressing the connexion of the index of 

 any one ray for a given medium, with three others supposed 

 assumed: thus successively each of the four remaining of the 

 seven standard rays have their indices found : or, in another 

 point of view, we may say, that (taking three terms of the 

 series) there are three constants to be found from observa- 

 tions. These depend on the medium ; when we eliminate 

 them therefore it is in effect equivalent to assuming three of 

 the indices given by observation. 



The very name of an approximate method commonly con- 

 veys the idea of a shorter and simpler process. In the pre- 

 sent instance, however, the case is quite otherwise. On look- 

 ing at the analysis contained in my former papers the exact 

 method might appear long and intricate, but in fact the pro- 

 cess of calculation is much simpler than might at first sight be 

 imagined, and shorter than in the approximate method. The 

 whole consists in the determination of the two constants log. 



