Method of Progressions. 327 



to one cf the most simple cases that occur in determining the 

 maxima and minima of quantities, let lis take the quantity 



ax — bx , in which x is variable, to find the value of x when 



ax —Ox IS a maximum. 



Consider x to be divided into in parts ; then, in the 7n—V^ 

 term, j:\vill be diminished by one of these parts 5 and in the 



7?i+ 1'^ term it will be increased by one of them. Consequently 

 we shall have 



and, S+T.ne™=<i^-'y-K^')"! 

 which being made equal, and reduced, give 



(i —W (X -1)'" 



/■! + — -.-^.ra + &c. >. 

 or, X = ■ X I 1 : 1^ ; 1 , 



rr^/p \ 1 1 n + r-l n + r — 2 —l J 



n + >o vi4 ^ — .— ^ — .m + &c.y 



But, in making the variation from the mP^ term the same at the 



m—V^ as at the m-\-V^ term, a cause of error is introduced, 

 which will be compensated if m be taken of such a value that 

 omitting the latter part of the above expression will counterba- 

 lance it. Then we shall have x''= =^7, when the expression 



n + r 



ax — bx is a maximum. 



This is true whether the indices be integers or fractions ; ne- 

 gative or positive; it is essentially the same as the fornmla derivetl 

 from the method of fluxions ; and is an example of the application 

 of the method of progressions to what is usually done by the di- 

 rect method of fluxions. 



I am, sir, yours, &:c. 



2, Grove Terrace, Lisson Grove, ThoMAS TreDGOLD. 



May 7, 1821. 



LV. Stridur s 



