8

n

"The letter S may be taken as an abbreviation of the word sum, thus Smam will represent the sum of n terms of a series of which the mth is am. It follows from this that the symbol S, Sam, will correctly represent the sum of a series Sm consisting of r terms, of which the mth term is the series Šam,n; and the same notation may be extended to any number of symbols of summation.”

A method analogous to the preceding has been devised by Mr. Jarrett for expressing by means of brackets the relation that exists between the different parts of a formula, when they are not connected either by addition or multiplication. An index is placed over the first bracket to denote the number of parts of which an expression consists, and a second bracket is so placed as to exhibit the connexion between two consecutive parts: thus

The brackets may be rejected after the expansion, whenever they are found not to possess any analytical signification; as in the preceding example.

In many investigations, particularly those connected with the theory of equations, it will be found convenient to denote by a symbol the sum of all the combinations of n quantities taken m at a time. Let therefore the letter C be taken as an abbre

m,n

viation of the word combination, then C.(a) may be used to denote the sum of all possible combinations of n quantities of which the 7th is a,, and of which m are to be taken at a time.

n

... In page 294 will be found the application of another symbol which has been devised by Mr. Jarrett to express what may be termed the symmetrical combinations of quantities, and which may be thus explained. Let there be two series a, a, &c. an b1, b2, &c. b2, each consisting of n terms, and let every possible combination be formed out of the first series taken m at a time; then if each combination be multiplied by n-m terms of the second series, such that those indices subscript which did not occur amongst the terms of the first series, may be found amongst those of the second, the sum of all the combinations

m,n-m

thus formed may be denoted by C, (a,b). This will probably be best understood by an example:

Some general theorems relating to these symbols will now be given, by means of which various analytical operations may be considerably curtailed. The demonstrations of some of these theorems will be found in the Paper by Mr. Jarrett above referred to the others may be demonstrated in a similar

manner.

[1] (1) Ÿm(am)=Ïm(an-m+1) ;

by this theorem the order of factors in a given factorial may be inverted.

(2) P(am)=P(am).Pm (ar+m);

by which any number of factors may be separated from the

m-1

m-1

a2m-2=a.P,(c2-2), and am-1a1.P,(C2−1)·

(5) If b is independent of m, then

n

=

P(ab)=b.P(4).

m

(6) P(a)=1, whatever may be the form of a„

The following formulæ very frequently occur; they may be considered as particular cases of the preceding:

(2) To divide a given series into two parts;

m

(3) To separate the even and odd terms of a series;

(6) If r is independent of n, and s of m, then

(9)

...

m

Sm S nam-n+1,n⋅

S,(,, ...m) • Sm ̧¤(m1, m2,

m1 m,-1

19

=Sm2Sm ̧••• Sm ̧p(m ̧ — m1⁄2 + 1,...m ̧-1 — m+1,m).

If ɑm+1—ɑm=bm, then

n

an+1=α1 + Smbm.

(10) If am+1+am=bm, then

(−1)n+1 an+1= —
'an+1 = − α1 + Ŝm (− 1)m−1 ̧bm2

m n

2m-1 n

m-1 n

a2n−r+1,r)·

2m-1

(14) S2 Sanr=S, S, (a1⁄2n-r ̧r + ɑ2-r+1,r) + S,ɑ2m−1,r •

[4] (1)

(2)

Cr,s (a, b)=an+1⋅ Cr,s (α, ·bs)+bn+1 · Cr,s (α, .b.).

*

Of the particular theorems which will here be considered, the first, and probably the most important, is the binomial theorem; the true development of which as first given by Mr. Swinburne and the Rev. T. Tylecote, and as more compendiously demonstrated by Mr. Jarrett,+ will afford a solution of all those contradictory results which have insinuated themselves into modern analysis, in consequence of neglecting the remainders in the development of diverging series: to obviate which, some writers have proposed a distinction between the mathematical signification of the term equality, and that which is implied in the ordinary acceptation of the

*The true development of the Binomial Theorem, Cambridge, 1827.

+ Cambridge Philosophical Transactions, Vol. III.