finding the Value of an infinite Series. 193 
t=£- 4C+6*/-4£+/, or d iv ; and v =fr- $c+iod-ioe 
+ $f-g, or d v ; and fo on of the following co-efficients, 
to wit, that every new co-efficient of the affumed feries 
is equal to the firft difference of the next order of the 
differences derived from the original co-efficients 
b, c, d, e, f gy by &.c. And from hence I concluded that 
the feries a- bx + cxx-dx^-rex^-fx^+gx^-hx 1 + See. 
was equal to the feries, 
a- 
bx 
T>'xx 
D 11 * 3 
> 1 1 1 r 4 
D 1 ^ 
Dv X 
D VI Y 7 
1 + x I+A> 
1+*' 
+ I + A 
I 
I 
Sec. 
Art. 6. The thought of fuppofing the original feries 
a-bx + exx-dx^ + ex^-fx^-r Sec. to be equal to the feries 
QX R XX S # 3 T X^ V X s 
‘ &c. containing the 
> 1 + # 
-M 
f+H 4 I +a] 
powers of x multiplied into the fame powers of the 
fraction ppp in order to accelerate their convergency, 
occurred to me in confequence of reading the late Mr. 
thomas simpson’s Mathematical Differtations, p. 62, 
63. concerning the fummation of feriefes, in which he 
makes a fuppofition of a limilar kind. Yet there feems 
to be a confiderable difference between his propofftion 
and that which is the fubjedt of thefe pages ; for he feems 
to fuppofe his quantities/), q, r, s, t, See. (which anfwer 
to a, by Cy dy £y Sec. in the notation made ufe of in the 
above feriefes) to form an increafing progreffion of 
terms, and accordingly fubtradts p from q, and q from r, 
Vol. LXVII. C c and 
