1 88 Mr. maseres’s Method of 
numbers a, h, c, d, e, f g, h> 8cc. fhall alfo form a de- 
er eafing progreffion ; and in like manner, that the dif- 
ferences of the faid third differences, or the fourth dif- 
ferences^ of the original numbers a , b, c, d, e,f g, h> 8ec. 
and the fifth and fixth differences, and all higher differ- 
ences, of the fame numbers, fir all alfo form decreafing 
progreflions. 
And 3dly, let x be a quantity of any magnitude not 
greater than unity. 
Upon thefe fuppofitions the value of the infinite feries 
a— bx.+cxx-dx' i - 1 rex*-fx i +gx ( ‘—hx 1 + 8 cc. (in which the 
fecond, fourth, fixth, and eighth, and every following 
even term, is marked with the fign - , or is to be fub- 
trafted from that which immediately precedes it) may be 
determined in the following manner. 
Art. 2. Compute the firlf, fecond, third, fourth, 
and other fubfequent differences of the co-efficients of 
the powers of x in this feries, that is, of the numbers 
b, c } d,.e,fyg r h, See. as far as fhall be convenient, Thefe 
differences will be as follows., 
Firft differences, b — c^c — d r d — e, e—f f—g, g — h, &c, 
Second differences, 
l — c — ( c —d y c — d—[d~eX, d~—e — e'—f~\f~g [ , f—g — Q— b\ &c* 
or, b—2 c-\- d y c—2 dr f e, d—z e + /,.. e—zf+g y f—zg -f b> &c. 
Third differences, 
c-\- d * {c^zd^t\ , c — 2 d-\- e — [d — ze -J-/1, d—2e±f — |V — 2 f+g\, 
e-zf+g—lf—zg &c._ 
Ol y 
