In all of which cases it is to be observed, that A, B, C, &c. denote the terms immediately preceding those in which they are first found,

Series of figurate numbers.

Hence by, putting m=1, 2, 3, 4, &c. successively, we shall have

1 + 1 + 1 + 1 + 1 + &c. =n

And, by putting m=1, 2, 3, &c. successively,

we shall have

1 +3x+6x2+10x3 +15x2 + 21x3 &c.=

(1 − x)2
1

1+4x+10x2 + 20x +35x2 + 56x3 &c. =

(1-x)3

1

(1-x)

1+5x+15x+35x3 +70x2 + 126x*&c. =

1+6x+21x2 + 56x3 + 126x* + 252x* &c.

1

(1-x)5

+

+

1.2.3.4

m(m + 1) │m(m + 1)(m + 2) ' m(m + 1)(m + 2)(m + 3)

&c.

And by putting m=1, 2, 3, &c. successively, we

+

1

+

1

+

1

n(n + 1)' (n + 1)(n + 2) ' (n + 2) (n + 3) ' (n + 3) (n + 4)

&c.

1 + x (1 + x)?

a+bx + cx2 + dx3

Σ=a+

D"x3

(1 + x)3 1 + x4

&c.

+ exa + ƒx3 + &c.

D"x3

D'x2

+ &c.

bx
+
1- -X (1-x) (1 − x3) ・ (1 − x1)

Where D', D", D"", &c. are the first terms of the several orders of differences of a, b, c, d, &c.;

That isn'b-c,D"=b-2c+d,D""b-3c+3d-e, D=b-4c+6d-4e+f, D'= b - 5c+10d - 10e + 5f-g, &c.

Binomial series, the terms of which are respectively multiplied by a, B, y, d, &c.

aam + Bbam-1x+ycam-2x2 + ddam-3x3 &c.

m-1

m-2

= a(a + x)TM + D'bx(a + x)TM-1 + D′′cx2(a + x)TM-2 + D'"'x3 (a + x)TM-3 &c.

m-3

Where D', D", D"", &c. denote the first terms of the several orders of differences of the quantities a, B, y, &c. as before.

And if p, p+q, p+2q, p+3q, &c. be put a, B, y, &c. we shall have

M-2

paTM + (p + q) bam-1x + (p + 2q) cam-2x2 +
(p+3q)da-3x3 &c.

s=p(a + x)TM + qbx(a + x)TM-1

for

(x − 1)2 ' (x − 1)3 ↑ (x − 1)+

&c.

Binomial series, the terms of which are divided by factors in arithmetical progression.