a + x : b + x> or <a: b, according as a < or > b.

a+ba-b: c+d: c-d;

and generally, ma + nb: pa + qb :: mc+nd: pc + qd.

(W. 171-182; E. 461-5.)

If a b c d,

and e: fg: h;

then

ae bf cg: dh.

(W. 184-8.)

If two numbers be prime to each other they are the least

in that proportion.

Compound proportion.

(W. 189.)

(E. 488-504.)

ARITHMETICAL PROGRESSION.

(18.) Let a represent the mth term of an arithmetical series,

m

Given the pth and qth terms, to find the 7th term.

(W. 212; E. 402-24; Bour. 193-8.)

(19.) Any three of the quantities a1, a, b, n, s; being given, the others may be found from the following formulæ :

(20.) The reciprocals of quantities in arithmetical progression are in harmonical progression, and conversely.

To insert n harmonical means between c, and e.

Given the pth and qth terms, to find the 7th term.

FIGURATE, AND POLYGONAL NUMBERS.

(21.) The nth term of the mth order of figurate numbers the sum of n terms of the (m-1)th order

n+- m. n + m 1.... n+1. n

=

a The above proposition may be thus compendiously stated:

For an explanation of this notation see the Appendix.

The nth term of the series of m-gonal numbers is the sum of n terms of the arithmetical progression

1, 1+ m −2, 1+2.m−2, &c.

The sum of the reciprocals of n figurate numbers of the mth

order

1.2.3...m

=

1

m−1 (1.2.3...(m-1)

1

(n + 1) (n + 2)...(n+m— 1)

(G. A. 132, 3; L. C. 91, 2; E. 425-39; B. 27, 8.)

To insert n geometrical means between c and e.

(W. 214-22; E. 505-22; Bour. 199-208.)