104 



ih 

 then differences are found by the same operations, and are 



a— ?t6+"-(''— "c "•(''-')■("— ^) ry+&c, 6— ?tc +"("— ') r/ ''■( »-i).('i— 2) p , 

 1T2 1.2.3 1 . 2 ~ 1 . 2 . 3 



<!v:r, &c. in which the successive quantities in the series of 

 powers are made also to enter into the exjiression for the dif- 

 ferences of the powers. The method of Fluxions, although 

 a particular case of the method of differences, cannot from 

 the above Formula be immediately deduced, since those 

 Formulae suppose the successive quantities to be known : 

 whereas, in the method of Fluxions, no account is to be taken 

 of the successive values of variable quantities, and to express 

 the several orders of Fluxions of powers, we have no other 

 notation than by expressing them in terms of the Fluxions of 

 the Roots. Therefore, putting Q instead of a—b, and R for 

 a — 2b+c, and, for the first of the third differences, or a— 3i-t- 

 tic — d putting S, &c. &c., if we would represent in terms of 



these the n differences of the powers, or such parts of thenj 

 as are constant, we would enlarge the analogy which has been 

 observed, in some cases, to hold between differences and 

 Fluxions. And this is the object of the following Essay, in 

 which the 5th Proposition and the differential Problems will 

 include the very useful and general Formula, for the co-effi- 

 cients in the method of finding Fluxions per Saltum,as disco- 

 vered by the Rev. Dr. Brinkley, the Professor of Astronomy, 

 in the University of Dublin, See the 7th vol. of " The 

 Transactions of the Royal Irish Academy," p. 327. 



ESSAY 



