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If c is a prime number + 4n+1, and a an odd divisor of 202 + cy', then R(ał(0-1) = c)= +1; + or -, according as a '#4n+1: if c4n +3,

R(alle-1) = c)=1. If c4n+1, and a is a divisor + en +1, or 8n +3, and b a divisor 8n +5, or 8n +7, of the formula x2 + 2cy; or if c#4n+3, and a son +1, 68n+3, then

R(at(c-1) = c)=1, and R(311-1) = c)=-1. No prime divisor of t + au can be expressed by more than one quadratic divisor of that form. (Leg. 148-99, 232.)

Properties of the quadratic divisors of t + au, a being a prime pumber #8n + 1.

(Leg. 372-80.) Every quadratic divisor of t + au contains at least one number prime to a, or to La, and <a. (Leg. 4105.)

Further investigation of the linear and quadratic divisors of f + 'cu'.

(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.)

TERNARY DIVISORS. (76.) No number *4n, or en+7, can be of a ternary form.

If c#4n+1, or 4n + 2, the formula t + cuo has at least one ternary divisor.

If c#8n +3, the formula t+cu’ has at least one quadratic divisor a, such that either a, or 2a is of a ternary form.

(Leg. 260_6.) If a quadratic divisor of t +cu' can be decomposed into

(a,y+b,x) + (azy+b,x) + (azy + b38)", then c=(a, b, - a,b)e +(a, b, - a,b,)+ (a, b, - a,b).

N

To find the ternary divisor corresponding to

c=F+ Go + HP Let F, G, H, be respectively resolved into fk, k,, gk,kg, hk, k3,

fk, and gkg, fk, and hkz, gk, and hką, being respectively prime to each other. Assume

a, b, - a,b,=fi fa+gaz + haz=0,

fb, +gb, + hbg=0; the required divisor is

k,(a,y + b *)] + k,(a,y+b*x)] + k, (azy+b3%). Since ky(azy +bx) + k, (azy +b,x) + kz (azy + b2x)=0, this divisor may be reduced by elimination to the form

ay? + byx +ex. 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 + au?; [2] the formula t + au will have as many ternary divisors as there are ternary forms of the number c; [3] each ternary divisor of t + au' has only one ternary form.

If the number a is comprised in a ternary divisor to +cu”, c will be comprised in a ternary divisor of t +au’; and the corresponding ternary values of a and c will be the same.

If py+299% +rx', a quadratic divisor of t +cu”, has several ternary forms, and in these forms given numbers are substituted for w 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 t + pu', c will also be a divisor t + bu, b being any number contained in the formula pyé + 294x + rx.

Properties of reciprocal divisors. (Leg.278—313.)

SCALES OF NOTATION.

(77.) Every number may be represented by

angol + an-por--+ an-271–2 + &c. + ayr + a, the radix r being any number, and any &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 2", if the number represented by the n last digits 2".

If the dimensions of any figure be expressed in feet, inches, and Ith 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.

Every number <2n+1 is the sum of some terms of the series

1, 2, 22, 23, &c. 2".

و

Every number <3n+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 (1005);

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°..

B' =

(2.) To reduce as bc" to the English scale.
A°=integral part of (as b'c" – 0,1 x as b'c"),

- integral part of 0,6 (preceding remainder),
C"= integral part of 0,6 (last remainder).
.18 = 54'.

1°=16 11' 11",i.
1' = 32",4.

l'=l' 85",185. 1"=0",324.

1"=3",08641 &c. The arc equal in length to the radius = 57° 17' 44' 48'" &c.= 57,22577 &c. = 206265" nearly. =635,66197 &c. (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. (sin a)2 + (cosa)=1. cosec a.sin a=1.

1 + (tan a)? =(sec a). = tan a.

1 + (cot a)? =(cosec a)?.

vers a=1- cos a. =cot a.

rs a=1-sin a. tan a.cot a =

suvers a=l+ cos a. =1.

(L. 35.)

sin a

cos a

COS a

sin a

covers

=l.

sec a.cos a=

(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 2nt=0,

cos 2n=1, tan 2n=0,

cot 2n=00, sec 2n=l,

cosec 2n=00. sin (2n +1)==1, cos (2n+1)==0, tan (2n+)=,

cot (2n + ?)=0,
sec (2n + ])==0, cosec (2n+)=1.
sin (2n + 1) = 0,

cos (2n +1)= -1,
tan (2n +1)=0, cot (2n +1)= -0,
sec (2n +1)= -1, cosec (2n +1)=00.
sin (2n +)=-1, cos (2n +*)=0,
tan (2n+*)*=-00 cot (2n +)*=0,
sec (2n+x)==0, cosec (2n +})==-1.

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