^3 a Dr. Rotheram onfome Properties 



But Arr=<2 — ar, whence at * =<2* — la ^r+a'- r"" j 

 and confequendy, I? — /?r'^=a^ — 2^^r+«*r* . 



Whence, after proper redudlion, r= 7 . 



This value of r being fubftituted in the equa- 

 tion x=a — ar, gives, after proper redudlion, 



x= ^ , ' Whence, x and r bein^ found, the 



feries will be known. 



Example. Let ^ = 24, and ^=1923 then 



a"^ — b 06 — 192 184 , , 2a^ 



r= ,=^ — : — - =^—1=1.; and x=~ — , 



a*-{-if 576 + 192 768 a^'+lf 



24X192X2 J .u r • • ^ 



•z-I -— =12; and the feries is 12, 6, 3, i^, 



768 



I, 1, -rV> &c. ad inf. 



Again. Let ^ = 243, and 3 = 295244:; then 

 y=^, and ;<'_i62; and the feries will be 162, 

 54, 18, 6, 2, y, ^, TT, &c. ad infinitum. 



In the infinite feries x+xr+xr"" +xr\ &c, : 

 inf. =«, a ^, becaufe, x=.a — ar, as abpve. 



By a fimilar deduftion, b = For the fame 



•' 1 — r* 



reafon, if xr" be the firft term of an infinite 

 feries (« being any pofitive whole number) then, 



,= !1I1. and i = ''llZ. 

 I— r I— r* 



The finite feries x+xr+xr^ +xr^ + ... to 



jcfti—j is the difference of two infinite feries, of 



which X is the firft term of the greater, and xr" 



of the lefs feries, and confequently the fum of 



the 



