m+3

S=

(a + x)TM÷3 — (¿1+3 — (m + 3)aTM+2x − +(m + 3)(m + 2)aTM+1y®

(m + 1)(m + 2)(m + 3)x3

Where the law of continuation of any series of this kind is evident.

Binomial series, the terms of which are respectively divided by the terms of a series of figurate

From which may be derived a number of particular series, by expounding r, by 1, 2, 3, &c. respectively.

2 1

1

4

\n' + &c.)

+ +2n2 + n° + &c.)

1

1

Log. ("+"') = (22, 1 + 3(2a + 1)2 + 3(2n+1}3 +

M 2n +

1 &c.)

7(2n+1)?

From which formulæ we can readily obtain the following logarithms:

To which there

32n + 5 2n + Log. (n-2)=

1

1

may

3 2n

5 2n

also be added the following: Log, n=

¦ ¦ (n − n ̄') — ¦ ¦ (n − n ') '+' (n−n^')' — ' (n−n^^)'&c. }

M

3

Log. n=

{(n−1) — ¦ (vn−1)'+' (Vn−1)' — ¦(vn−1)'&c.}

3

Log. n=

Where r denotes any root whatever of n; and consequently may be so taken that the value of Vn, found by repeated extractions of the square root, shall differ but little from 1.

The following series may also be used for determining the number answering to a given loga

In all of which cases м must be taken = 1, for hyperbolic logarithms, and 2.302585093, for commmon logarithms.

Where e number where hyperbolic or Neperian logarithm is 1, which is known to be 2.7182818284.

MISCELLANEOUS QUESTIONS.

1. A person being asked what o'clock it was, answered, that it was between eight and nine, and that the hour and minute hands were exactly together; what was the time?

Ans. Sh. 43min. 38 sec. 2. A certain number, consisting of two places of figures, is equal to the difference of the squares of its digits, and if 36 be added to it the digits will be inverted; what is the number? Ans. 48

3. It is required to find three numbers such, that

shall be all equal to each other.

Ans. 280, 294, and 300 4. What two numbers are those, whose difference, sum, and product, are to each other as the numbers 2, 3, and 5, respectively? Ans. 2 and 10

5. A person, in a party at cards, betted three shillings to two upon every deal, and after twenty deals found he had gained five shillings; how many deals did he win? Ans. 13

6. A person wishing to enclose a piece of ground with palisadoes, found, if he set them a foot asunder, that he should have too few by 150, but if he set them a yard asunder he should have too many by 70; how many had he? Ans. 180

7. A cistern will be filled by two cocks, A and B, running together, in twelve hours, and by the cock A alone in twenty hours; in what time will it be filled by the cock B alone? Ans. 30 hours 8. Given (x+1) × (x2 + 1) × (x3 +1)=30x3, to find the value of r by a quadratic equation.

Ans. x=(35) 9. Required that arithmetical progression, whose number of terms is 10, sum of the terms 185, and sum of the cubes of the terms 104525.

Ans. 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 10. A, B, and c, are to share 100000l. between

1 1 3' 4'

them, in the proportion of and, respectively;

but c's part being lost, by death, it is required to divide the whole sum properly between the other two.

Aus. A's share 5714237. and B's 42857-37,1.

11. If three agents, A, B, C, can produce the effects a, b, c, in the times e, f, g, respectively; in what time would they jointly produce the effect d.

12. What number is that, which being severally added to 3, 19, and 51, shall make the results in geometrical progression? Ans. 13 13. It is required to find two mean proportionals between 3 and 24; and four mean proportionals between 3 and 96.

Ans. 6 and 12; and 6, 12, 24, and 48. 14. It is required to find six numbers in geometrical progression such, that their sum shall be 315, and the sum of the two extremes 165.

Ans. 5, 10, 20, 40, 80, and 160 15. The sum of two numbers is a, and the sum of their reciprocals is b; required the numbers.

16. The sum of three numbers in harmonical proportion is 13, and their continued product is 72; what are the numbers? Ans. 6, 4, and 3

17. A certain number of men were employed in making a fish pond, and when they had been at work 24 days, and had half finished it, 16 men more were set on, by which the remaining half was completed in 16 days: how many men were employed at first; and what was the whole expence, at 1s. 6d. a day per man?

Ans. 32 men; and the whole expence 115l. 4s. 18. It is required to find two numbers such, that if the square of the first be added to the second, the