finding the Value of an infinite Series , 8 cc. 189 
or, b — 3^ + 3^ — e> c—^d+ze—f, d-ze+zf—g, e— 3f+3g~ b> &c. 
Fourth differences, b — 3 c 4 3 */ — e — \c — 3 d + 3 *—/), 
3 -^+ 3 e ~f— + , 
3 * + 3f—g— 3/ + 3£ — ^ , &c. 
or, £— 4^4-6/— 4*4-/, c— - 4d+6e — 4/4-^, ^—4*4-6/— 4^+^, &e Q 
Fifth differences, £ — 4 c + 6d — 4*4*/ — |r— 4 d+ 6 * — 4/4-~g| , 
c— 4^+6^ — 4/+ £ — [d—^e 4- 6f—~4g 4^\ , &c. 
or, ^—5^4-i°^— 1°^4-5/~ g, c— $d+iOe — lof+sg — b , &c* 
Sixth differences, 
^—5^4-10^—10^4-5/—^ — !r— 5^4- io<?— 10/4- $g — h\ > &c 9 
or, b — 6 r 4 -i 5 ^ — 20^4- 15/— -6^4-^, &c. 
Let the firft difference of the firft order, to. wit, 
b.—c , be called d 1 ; 
and the firft difference of the fecond order, to wit, 
b—zc+d, be called n”; 
and the firft difference of the third order, to wit, 
b — 3 c 3 d—e y be called b 1H ; 
and the firft difference of the fourth order, to wit, 
b~ 4C+ 6 d— 4 -e+f, be called d 1v ; 
and the firft difference of the fifth order, to wit, 
b— $c+iad—- ioe+ $f—g, be called d v ; 
and the firft difference of the fixth order, to wit,,. 
b- 6 c+ 15 d—zoe+ 1 5/- 6g+h, be called d vi ; 
and in like manner let the firft differences of the feventh, 
eighth, ninth, and tenth, and every following order of' 
differences be denoted by d vii ,.d vui , n IX , d x , Sec. that is, , 
by the capital letter d, with a Roman numeral figure 
annexed 
