196 
Mr. maseres’s Method of 
feries 1 - 
tt 
3rr 5 
>]r 6 ^ 9 r 
1 1 r 
13;- 
1 5 r ,+ 
Sec. and 
tt 
muft fubftitute x inftead of — in the terms of this laft fe- 
rr 
ries, by which means it will be converted into the feries 
X XX 
I -T “ 
3 5 
+ — + JZ --—r+ Sec. This feries is of 
the fame form with the original feries above-mentioned, 
a-bx + cxx—dx i + ex^-fx s +gx (, -bx lj r Sec. the numeral 
co-efficients 1 , }, j, fr, T V, 8cc. of the powers of a; 
in the former feries anfwering to the literal or general 
co-efficients a, b , c, d, e, f \ g, b, Sec. of the fame powers in 
the latter feries. Andthefe numeral co-efficients evidently 
form a decreafing progreffion, as the co-efficients a, b, c, 
d, e y f, g, b, Sec. are fuppofed to do; and we fhall find, 
upon examination, that the differences of thefe numeral 
co-efficients, of the feveral fuccefiive orders, alfo confti- 
tute decreafing progreffions, as the feveral fuccefiive or- 
ders of differences of the co-efficients a , b y c , d, e, f, g, b, 
Sec. are fuppofed to do. Confequently the feries 
X XX X 3 X 4 X ^ X ^ X ^ 
1 ~ T + T~ 7’ f 9 - rT + T3~r^ + w ill he e fi u£ fi to the dif- 
ferential feries 
bx 
T > 1 X X D 11 * 3 D 111 * 4 D IV * 5 
D v / 
D y l X 7 
See. if 
1 +*1 i+*r 1 1 +n 5 1 +^1° 1 -(-a) 
we fuppofe the letters a, b, c, d, e, f, g, b, Sec. to be 
equal to the numbers 1, }, j, }, f r , T ' T , T ' T , &c. and 
D ', D n j D in , D ,v , d v , d vi , 8cc. to be the firft differences of 
4 the 
