262 Prof. Powell's Observations on some Points 



theory, it has been brought to bear on the question of the 

 dispersion. Yet there are still those who are so unreasonable 

 as to inveigh against the theory,because it has not, in that com- 

 paratively short period, succeeded in clearing the subject of all 

 its difficulties ; and who, instead of any degree of satisfaction at 

 what has been done, express only displeasure at what has not 

 been effected. Passages might be cited (were it to any pur- 

 pose to do so) from the writings of some philosophers of the 

 present day, in which complaints of this nature occur, in- 

 volving remarks and reflections which we cannot well set 

 down to ignorance, and which, in any other point of view, 

 can neither be regarded as peculiarly creditable to the taste 

 of the writers, nor to their appreciation of what is justly due 

 to a series of researches of such a character as must fairly be 

 allowed to belong to those in question, even if they could be 

 shown to have failed in their object. 



It is not however my design in the present observations to 

 enter into controversy, but merely to offer a few remarks on 

 one or two points connected with that part of the theory 

 which at present seems most open to difficulty and objection ; 

 and to which the attention of those who are in a condition to 

 grapple with the difficulties of this intricate portion of dy- 

 namical research, is at present most powerfully called. 



In my first deductions from the theory of M. Cauchy, and 

 my earliest calculations to compare it with observation, I 

 made use of a formula which was allowed all along to be but 

 an imperfect and approximate one, and which was applied 

 numerically only by the aid of an indirect and tentative pro- 

 cess. A direct mode of calculation by it, was indeed stated and 

 explained (under a certain material limitation) (see London 

 and EdinburghJournalof Science, April 1836, vol.viii. p. 309), 

 but this was in practice not less troublesome than the former. 



The other and more exact methods proposed by Sir W. R. 

 Hamilton and Mr. Kelland, by which my later computations 

 were carried on, involve a greater number of constant coeffi- 

 cients; which by consequence, directly or indirectly, must 

 be assumed from observation. The nature of these assump- 

 tions, and the precise difference between the two formulas, 

 I pointed out in a paper in the Lond. and Edinb. Journal of 

 Science, January 1836, (p. 26,) and March J 836, (p. 206). 

 It will be desirable here briefly to restate them : 



The exact formula deduced immediately from M. Cauchy's 

 theory is, f 





This, it may be well to observe, is equivalent to the sum 



