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If c is a prime number 4n+1, and a an odd divisor of

x2+cy, then

R(a(1)÷c)= ±1;

+ or -, according as a 4n+1: if c4n+3,

R(a(-1)+c)=1.

If c4n+1, and a is a divisor 8n +1, or 8n +3, and b a divisor 8n +5, or 8n +7, of the formula x2+2cy2; or if c4n+3, and a‡8n±1, b‡8n±3, then

R(ał(1)÷c)=1, and R(61)÷c) — — 1.

=

No prime divisor of t+au2 can be expressed by more than one quadratic divisor of that form. (Leg. 148-99, 232.) Properties of the quadratic divisors of t+au3, a being

a prime number 8n+1.

Every quadratic divisor of t+au2 number prime to a, or to a, and <a.

(Leg. 372-80.) contains at least one (Leg. 410-5.)

Further investigation of the linear and quadratic divisors

of t+cu2.

(Leg. 204-12.)

Tables of linear and quadratic divisors explained.

(Leg. 213-25.)

Application of the above tables to determine

[1] a prime number greater than a given number;

[2] whether a given number is prime or not. (Leg. 246—59.)

(76.) No number

TERNARY DIVISORS.

4n, or 8n +7, can be of a ternary form. If c4n+1, or 4n+2, the formula t+cu2 has at least

one ternary divisor.

If c8n+3, the formula ratic divisor a, such that either a,

+ cu2 has at least one quador 2a is of a ternary form.

(Leg. 2606.)

If a quadratic divisor of t2 + cu2 can be decomposed into (a1y+b1≈)2 + (a ̧y + b2x)2 + (α ̧y+b3x)o,

then c=

= (a ̧b ̧ — a ̧b ̧)2 + (a‚b‚— a‚b‚)2 + (a ̧b ̧—à ̧b2)°.

N

To find the ternary divisor corresponding to

c=F2+G2+H2:

Let F, G, H, be respectively resolved into fkk2, gk ̧k ̧, hk2k ̧, fk, and gk,, ƒk, and hk,, gk, and hk,

being respectively prime to each other.

Assume

fa1+ga2+ha,=0,

ƒb1+gb2+hb ̧=0;

the required divisor is

2

2

k ̧(a ̧y+b ̧×) | ̊ +k2(a ̧y+b2x) |2 + k ̧(α3y+b3x) |2

2

3.

Since k, (a,y+b ̧≈) + k2 (α2y+b2≈)+k ̧(α ̧y+b ̧≈)=0, this divisor may be reduced by elimination to the form

ay2+ byx+ex2.

If a=e, b a or e, or e=0, and at the same the least of the quantities a and e>2, this divisor will have two ternary forms, and no more: if these conditions are not fulfilled, there will be only one ternary form, corresponding to any given value of c.

If either c, or c, is a prime number,

[1] two different ternary forms of c cannot correspond to the same ternary divisor of t+au2;

[2] the formula + au2 will have as many ternary divisors as there are ternary forms of the number c;

[3] each ternary divisor of 2+au has only one ternary form.

If the number a is comprised in a ternary divisor ť2 +cu2, c will be comprised in a ternary divisor of + au2; and the corresponding ternary values of a and c will be the same.

has

If py2+2qyx+rx2, a quadratic divisor of t+cu2, several ternary forms, and in these forms given numbers are substituted for x and y, the results will all be different, if each of them<c.

Suppose p prime to c; then if c is a divisor of t2 + pu3, e will also be a divisor 2+ bu, b being any number contained in the formula py2+2qyz+rz2.

Properties of reciprocal divisors.

SCALES OF NOTATION.

(77.) Every number may be represented by

1

-2

(Leg.278-313.)

ɑ ̧p2 + ɑn - 1 p2 - 1 + an − 21 + &c. + a1r + a,

the radix r being any number, and a„, &c. integers<r.

To transform a number from one scale to another: divide the number in its own scale by the new radix, and repeat the same process with that and all succeeding quotients; the remainders taken in order, beginning with the last, will be the digits which express the given number in the new scale.

In any scale, radix r, if the number itself and the sum of its digits be respectively divided by r-1, the remainders will be the same.

If the sums of the odd and of the even digits be severally divided by r+1, the difference of these remainders will equal the remainder of the number itself divided by r+1.

Any number

last digits 2".

2", if the number represented by the n

If the dimensions of any figure be expressed in feet, inches, and th parts, the area or solidity may be most readily found by transforming the number of feet into the duodenary scale, and multiplying together the results in that scale; observing the same rules as in the multiplication of decimals.

series

Every number<2"+1 is the sum of some terms of the 1, 2, 22, 23, &c. 2".

Every number <3"+1 may be expressed by the sums or differences of some terms of the series

1, 3, 32, 33, &c. 3".

(B. 121—9.)

100

TRIGONOMETRY.

(1.) Divisions of the circle. The circle has usually been divided into four quadrants;

each quadrant into 90 degrees, (90°);
each degree into 60 minutes, (60');

each minute into 60 seconds, (60′′).

The more minute divisions are generally expressed in decimal parts of 1"; they are however sometimes expressed by dividing each second into 60 thirds (60′′′′), and so on.

The French have recently divided the quadrant into 100 grades (100);

each grade into 100 minutes, (100');

each minute into 100 seconds, (100"); and so on.

In Astronomy the circle is sometimes divided into 12 signs, each of which contains 30°.

(2.) To reduce a b' c" to the English scale.

A° = integral part of (a2 b' c" — 0,1 × aa b' c"),

B' = integral part of 0,6 (preceding remainder),
C"integral part of 0,6 (last remainder).

[blocks in formation]

The arc equal in length to the radius

=57° 17′ 44′′ 48" &c.=57°,22577 &c. = 206265′′ nearly.

[blocks in formation]

(W. 19; L. 1-12; C. 631; Enc. Met. Vol. 1, p. 672.)

(3.) To adapt any formula in which radius is unity to the general radius, r: multiply each term by such a power of r as shall render it of the same dimensions as the highest mentioned in the given formula. (L. 24; W. 18.)

(4.) Mutual relations of the trigonometrical lines.

[blocks in formation]

(5.) The sine is positive in the first and second quadrants, and negative in the third, and fourth; the cosine is positive in the first and fourth, and negative in the second and third quadrants: from these data the signs of all other trigonometrical lines may be determined.

sin 2nπ=0,

tan 2nπ=0,

sec 2nπ= 1,

sin (2n+1)=1,
tan (2n+1)π= ∞,
sec (2n+1)π= ∞,
sin (2n+1)=0,
tan (2n+1)=0,
sec (2n+1)π= −1,

sin (2n+3)= −1,
tan (2n+23)π=
sec (2n+2)π= ∞,

3

cos 2nπ = 1,

cot 2nπ= ∞,
cosec 2n = ∞.

cos (2n+1)=0,
cot (2n+1)π=0,
cosec (2n+1)=1.
cos (2n+1)π=−1,
cot (2n+1)= ∞,
cosec (2n+1)=x.

cos (2n+2)π=0,
cot (2n+2)=0,

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