APPENDIX.

INVESTIGATION OF THE MAGNETIC CURVES.

I HAVE been favoured with the following investigation of the curves, to which a needle of indefinite minuteness will be a tangent, by Mr. Playfair, Professor of Mathematics in the University of Edinburgh.

Two magnetical poles being given in position, the force of each of which is supposed to be as the mth power of the distance from it reciprocally, it is required to find a curve, in any point of which a needle (indefinitely short) being placed, its direction, when at rest, may be a tangent to the

curve?

1. Let A and B (Plate IV. fig. 21.) be the poles of a magnet, C any point in the curve required; then we may suppose the one of these poles to act on the needle only by repulsion, and the other only by attraction, and the direction of the needle, when at rest, will be the diagonal of a parallelogram, the sides of which represent these forces. There fore, having joined AC and BC, let AD be drawn parallel to BC, and make

1

1

ACm BCm : AC: AD; join CD, then

CDF will touch the curve in C.

2. Hence an expression for AF may be obtained. For,

ACm+1

by the construction, AD = BC, and since BC: AD: :

BF: FA, and BC

AB X ACm+1 BCm+1 ACm+1°

AD: AD: AB: AF, we have AF

3. A fluxionary expression for AF may also be found in terms of the angles CAB, ABC. In CF take the indefinitely small part CH, draw AH, BH, and from C draw CL perpendicular to AH and CK to BH. Draw also BG and AM at right angles to FH. Let the angles CAB = 9, and CBA; then CAH, and CBH; also CL = AC x, and CK = — BC x Now HC: CL:: AC2 x and for the same reason BG = — HC

AC: AM=

Therefore since AF: FB:: AM: BG, AF:

BC2 X

HC

FB::

HC

, and AF: AB:: sin. 2

HC

į sin. q2 + è sin. 42'

; wherefore if ABα, AF =

4. If this value of AF be put equal to that already found, a fluxionary equation will be obtained, by the integration of which the curve may be constructed. Because AF AB × ACm+1

a sin.

BC m+1-AC m+1; and since AC sin. (+4)'

a sin.

=

sin. (+), we have by substitution AF

and BC

a sin. 42

sin. 2+ sin. 2. Hence,

42.

sin. 2 × sin. m+1 + ø sin. ↓ m+3=- sin. 2 × sin. ♦ m+1 + ♦ sin. ↓ m+3, and therefore į sin. 4m-1 —— ♦ m---1; and also, sin. 4m-1 + sin. 4 m-1 = C.

5. These fluents are easily found when m is any whole positive number.

If m = 1, we have +0.

The first of the above equations belongs to a segment of a circle described upon AB, which therefore would be the curve required if the magnetical force were inversely as the distances.

If the magnetical force be inversely as the square of the distance, that is, if m = 2, cos. + cos. is equal to a constant quantity. Hence if, beside the points A and B any other point be given in the curve, the whole may be described. For instance, let the point E (Plate IV. fig. 22.) be given in the curve, and in the line DE which bisects AB at right angles. Describe from the centre A a circle through E, viz. QER; then AD being the cosine of DAE to the radius AE, the sum of the cosines of X will be everywhere (to the same radius) = 2 AD = AB. Therefore to find E', the point in which any other line AN, making a given angle with AB, meets the curve, draw from N, the point in which it meets the circumference of the circle QER, NO, perpendicular to AB, so that AO may be the cosine of NAO, and from O toward A take OP = AB, then AP will be the cosine of the angle ABE'; so to find BE' draw PQ perpendicular to AP, meeting the circle in Q; join AQ, and draw BE' parallel to AQ, meeting AE' in E', the point E' is in the curve. In this way the other points of the curve may

be found.

The curve will pass through B, and will cut AB at an angle of which the cosine RB. If then E be such, that AEAB, the curve will cut AB at right angles. If E" be more remote from A, the curve will make with AB an obtuse angle toward D; in other cases it will make with it an acute angle.

A construction somewhat more expeditious may be had by describing the semicircle AFB, cutting AE in F, and AE' in N, and describing a circle round A, with the distance AL 2 AF, cutting AE' in b. If BG be applied in the semicircle AFB Nb, BG must cut AN in a point E' of the curve, because ANBG = 2 AF, and AN and GB are cosines of the angles at A and B.

As the lines AN and BG may be applied either above or below AB, there is another situation of their intersection E'. Thus An being applied above, and Bg below, the intersection is in e'. The curve has a branch extending below A; and if De be made = DE, and B e be drawn, it will be an assymptote to this branch There is a similar branch below B. But these portions of the curve evidently suppose an opposite direction of one of the two magnetic forces, and therefore have no connection with the position of the needle.

VARIATION OF THE COMPASS*.

THE variation of the Compass, is the deviation of the magnetic or mariner's needle from the meridian or true north and south line. On the continent it is called the declination of the magnetic needle; and this is a better term for reasons which will afterwards appear.

Our readers know, that the needle of a mariner's compass is a small magnet, exactly poised on its middle, and turning freely in a horizontal direction on a sharp point, so that it always arranges itself in the plane of the magnetic action.

About the time that the polarity of the magnet was first observed in Europe, whether originally, or as imported from China, the magnetic direction, both in Europe and in China, was nearly in the plane of the meridian. It was therefore an inestimable present to the mariner, giving him a sure direction in his course through the pathless ocean. But by the time that the European navigators had engaged in their adventurous voyages to far distant shores, the deviation of the

• It is necessary to remind the reader, that the following article on the VARIATION OF THE COMPASS, was published a considerable time before the article on MAGNETISM.-ED.