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(a, + a2 + az + ... + am)" (26.)

= the sum of all possible 1.2.3...n values of

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(1.2.3...rı) (1.2.3...r,) (1.2.3...rg)...... (1.2.3...rm)' subject to the condition that

7 +1, +7,+ +?m=n; the values of rı, r,, &c. being either 0, or a positive integer; and the partial divisor for a value 0 being 1.

(27.) (a + ax + Qow? + Q 3203 + &c.)"

n(n-1) an-1

an 2 1

1. 2

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+

n

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+1.2

+ Tan-la, n(n-1)(n − 2) an-sa for

? +

n(n-1)(n-2)(n − 3),n-4a* 3

1. 2 . 3 . 4 n(n-1)

n(n-1)(n-2)

an – 3a; a,
1. 1

1. 2. 1
n(n-1)

- an -2 a, a,
1. 1
n(n-1)

a-a
1. 2

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+

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n

+

03

+

+

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(Lacr. Introd. Diff. Calc. 19—24; Arbogast, Calc. des Deriv. 33.)

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- 3 12

Continuation of Note.

az

ai +

5 3 2 +

+

A.az + n 4 3 2

3
az

an
+
az +a, (

+ 02.04
2

5

an

saj

03

n

an-4

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ai

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an- 3

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n

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In this theorem, n has been considered a positive integer; if an expansion be required for any value of n, we must expand in a similar manner

m

m-1.a,"

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