II. On Algebraic Notation.

BY THOMAS JARRETT, B. A.

OF CATHARINE HALL.

[Read Nov. 12, 1827.]

1. THE object of this paper is to propose a notation by means of which the analysis of series may be facilitated, and to suggest a few modifications in the received system of algebraic notation.

2. The principle of the notation about to be proposed, though sufficiently obvious, and at the same time by no means novel, is one capable of being applied much more extensively than it has hitherto been. It is as follows: if a, is any function of m, and if A represent that operation by which any quantities are combined in a given manner, then the symbol 4" (a) may be used to represent the result of such combination applied to n functions, of which the mth is a. Again, if is a function of any quantities, one of which is x, then B.. (v) may represent the performance on of any operation in which a alone is considered as variable, and B”.(v) will equally represent the result of that operation when repeated n times.

3. Let then the letter P be taken as an abbreviation of the word product, and P.am will denote a product of n factors of P", which the mth is am.

4. PROP. To invert the order of the factors in a given

factorial,

order

5.

= An· An-1 · An-2. a1, taking the factors in an inverted

= Pm.an-m+1·

PROP. To separate any number of the factors from the rest,

= cm, m being any positive integer, then shall

·· am = a1.P ̧TM−1 (c,).

= cm, then a2m-2 = a。. Pm-1 (C2r_2),

and a2m-1 = a1. P‚m−1. (C2r−1).

These two examples will be found of great use in the investigation of the general term of many series.

6. PROP. If b is independent of m, then shall

Pm (a,b) = b2. P”m (am).

For, P. (a,b) = a ̧b.ab.a ̧b...ab

= br. а1 а2 A、... An

= br. P”m (am).

7. PROP. Whatever is the form of am, Pom · (am)=1.

For, P. (a) = P," +r. (a) = P". (am). P'm (am+n-r), (Art. 5.)

m

8. The most common form in which factorials occur, is that of an arithmetical series. A factorial of this kind consisting of m factors, of which n is the first, and of which the common difference is ±r, may be denoted by n ; the particular case

in which the common difference is

n,

m

-

m, tr

1, we may represent by

and if, in this case, mn, the index subscript may be

omitted:

9.

Thus n = n. (n±r) (n±2r) ... (n±m−1.r),

Let the letter S be taken as an abbreviation of the word sum, and S. am will represent the sum of n terms of a series of which the mth term is am.

= A2 + An−1 + An_2+ &c. + a1, by inverting the series

n

= Sman-m+1•

11. PROP. To divide a given series into two series.

(a1 + a2+a2 + &c. +a,) + (a,+1 + a,+2+&c. + an)

3

S.am+S". Ar+m

12. PROP. To separate the even and odd terms of a series,

13. THEOREM. If b is independent of m, then shall

m

14. The symbol S. Sam,n will correctly represent the sum of a series consisting of r terms, of which the mth term is the series Sam,n; and the same notation may be extended to any number of symbols of summation.

15. THEOREM. (Smam.) (S3n bn)

bn

For, (S'm am) (S„bn) = a1. Sεn b2 + a2 S3n bn + &c. + a,. S3, bn,

by actual multiplication,

shall

= Smam. Sn. by

n

16. THEOREM. If r is independent of n, and s of m, then

17. PROP. To arrange S". S",.am,r. x-1 according to powers

of x.

T

+ S3‚a3⁄4‚‚· x2-1 + &c. + S", . An,r. x2-1.
aз, 7.

Taking successively the 1st, 2d, and 3d, &c. terms of these different series, we get

'r

S.Sm, am, r.x-1

=

(a1,1 + a2,1+ag, 1 + &c. + an, 1)

+ x (α2, 2 + A3, 2 + A3,2 + &c. + ɑn, 2)

+ x2 (α3,3 + α4,3+A5, 3+ &c.+an, 3)+&c.+x^-1. ɑn, n)

-2

1

= S",. a,,1 + x . §,”−1 a‚ + 1, 2 + x2. §,n−2 α, +2,3 + &c.+x2¬1. An‚ n