40 



(III.) 



Integration of the Equation. 



r^ A- U — G + m-K cos (cn—'jr) = o 



tit- ' ^ ' 



G and K being constant quantities, and 



1. c= any number greater or less than unity ^ 



2. c=: 1 and rn^ of any magnitude. 



3. c= I and nf a very small quantity. 



The subsequent mode of integration will be more readily un- 

 derstood by a short ])refatory explanation.* 



Let us suppose we have in any manner arrived at an equation. 



d*"?) (x, y^ — o, M'henxz=o and rfa; is constant, m representing 

 any manher not less than n, we can then condude 

 (p [x, y)- ci + c^x + Cj.r- + c„ x"-' 



The equation d"' <p (x, y^ - o may be called the m"' particular 

 fluxion of tin's equation with reference to .r = o. 

 If we have only equations of particular fluxions commencing with 

 the order n, the values of Cj, c,, Cs, &c. are arbitrary. But if 

 there be preceding equations of particular fluxions not contained 



* The mode of integration liere shortly explained derives its advantage from the method 

 of finding fluxions per snltuni. Theorems for this purpose were given by me in a paper read 

 before this Academy in the year 1798, and publislicd in the Seventh Volume of the Trans- 

 actions. Tliese theorems were considerably extended, and applied both to the direct and in- 

 verse reduction of analytical functions in a work which I had prepared for the press, but hav- 

 ing adopted a notation dilFering both from the fluxional notation, and that of the differential 

 calculus, I was for many years deterred from publishing it. Lately, however, 1 have again 

 resumed the subject, changing the notation iuto the usual notation of the differential tllleulus, 

 and I hope soon to offer the result to the notice of mathematicians. The method of integra- 

 tion here used belongs to that division of the work entited " On the inverse reduction of 

 analytical functions." 



