68 PROFESSOR POWELL'S RESEARCHES TOWARDS ESTABLISHING 



soon received from him a statement of the results of theory for oil of cassia, in which 

 the discrepancies were almost wholly removed. 



I have since verified that calculation, and have performed similar computations 

 for the only other cases in which material differences before appeared. 



The object of the present communication is to state these results, with the necessary 

 data of the calculations ; and further, to elucidate the general method, so as to render 

 it more readily applicable to other cases which may arise in the further prosecution 

 of the determination of refractive indices ; and to notice the present condition in 

 which the theory may be considered to stand with respect to this material portion of 

 its experimental evidence. 



Explanation of the Formula. 



The formula adopted in my preceding papers includes essentially the development 

 of the term 



'■" (t) 



a)- 



This of course gives a series involving the even powers of X with certain coefficients. 

 And the practical differences in the methods of calculation turn entirely on the number 

 of terms to which it may be thought necessary to pursue this series, or the mode of 

 finding or eliminating the coefficients. 



As it does not enter into my present design to refer to the physical principles of 

 the theory, I will merely here observe, that though such principles have been assumed 

 under some difference of aspect by the several eminent mathematicians who have 

 treated the subject, yet the formulas deduced for the dispersion have, in every in- 

 stance, resulted the same as far as the /brm of the series is concerned, differing only 

 in regard to the nature of the summation and the coefficients involved. 



As it is in regard to the numerical comparison with experiment that I am at pre- 

 sent engaged in considering the subject, I have been chiefly interested in comparing 

 these methods so far as to see whether, when one might fail in giving sufficiently close 

 coincidences, another might cause the discrepancies to diminish or disappear. 



In this view then, referring to Mr. Kelland's method, it may be necessary for its 

 better elucidation to state it generally as follows. Supposing it sufficient to take 

 three terms of the series, the relation of the refractive index [uj) to the wave-length 

 in the medium {\) may be expressed thus : 



Our comparison, however, is to be made with the wave-length in air or vacuum, 

 which, in order to express that in the medium, must be reduced in the ratio of the 

 refraction for the medium and- for the ray, (which is not expressed in the author's 

 formulas,) or, X being the wave-length in air, we must take 



