magnetism, at least within the certain limits of time, that are more than sufficient for observations; it follows from this, that the substitution of the constant g' sin. i for the constant g', does not in any way change the conclusions that may be deduced from the experiments, and that we may draw the same results from the observation of the number of oscillations, by employing the horizontal instead of the dipping-needle, since the relations only are concerned, in which the absolute and constant intensity of the force arising from terrestrial magnetism disappears. See in this respect (pp. 182. and 187.) the application that we have made of the method of oscillations of the horizontal needle for the determination of the laws of distance, and the distribution of magnetic forces in a magnetised bar.

NOTE D. (p. 186.)

Demonstration of the Law of the Inverse of the Square of the Distance, by Calculation of Magnetic Curves.*

SETTING out from the form of magnetic curves observed by Dr. Roget, we are about to show that the action of each of the poles of a magnetised bar upon those which are determined by induction in a particle of iron filings, varies in inverse ratio with the square of their distance.

Let A and B represent the two poles of a magnet, of a contrary or of the same name, between which the magnetic curve is formed; and let us first seek the equation of the latter of these, in respect to two rectangular axes drawn in its plane. Let м represent any one of its points, whose co-ordinates will be x, y; let the line AB represent the axis of x, its origin being in the middle o, the positive a being calculated on the side oA, and the positive y on the side of the curve.

Let h represent the length OA=0B, r, r' the distances AM, BM, and i, i' the angles MAB, MBA; finally, let fall from м upon AB the perpendicular MP. We have, evidently, AP AM COS. MAB=7 cos. i; on the other hand, AP=AO-OP=h-x, consequently ; in like manner, we shall find cos. i'= =h+x. Ac

cos. i=

h-x

cording to Mr. Barlow's first law we have, in the whole extent of the curve, cos.icos. i'const., taking the upper or the lower

* This note was furnished to me by M. Ch. Cellerier; and is now pub. lished for the first time.

line, according as the poles A and B are of a contrary or of the same name. By substituting the above values for cos. i, cos. i', we shall find the equation of the curve,

Here r and are put in place of their values,

Now, let A'B' be the poles of a particle of filings situated at M; A' being of the same name as A. The resultant of the actions of A and B upon A', and that of their actions upon the very neighbouring point B', will evidently be two equal forces, but in contrary directions; their common direction is that which will be assumed by the particle A'B', if it is free to move. It will therefore be tangent to the curve formed by several particles, when adhering to each other, by poles of the contrary name; for the little right line A'B' is then an element of this curve.

Let g represent the angle comprised between the axis of x and the tangent to the curve at the point м. We shall clearly have,

dy

tang. 9-dx

dy dx

and the deduction will be obtained by dif

ferentialising the equation of the curve, making x and y vary together. We then, by the known rules, find

―rdx-(h-x)dr ̧ r'dx−(h+x)dr′
±

7.2

'2

=0;

by differentialising the values of rr, we further obtain.

dr=

-(h-x)dx+ydy, dr= (h+x)dx+ydy

r

дом

By substituting in the previous equation, and multiplying it by

r'3 (ydx+(h−x)dy)±r3 ((h+x)dy—ydx)=0;

y

dy

and the value that we deduce from or tang. g is

g=

tang. 9=73 (h-x) + r3 (h+x)

The tangent to the curve coincident with the direction of the resultant of the actions of A and B upon A', letting F represent this latter, and x, y its components according to the axes, we shall have

X=F cos. g; Y=F sin. g, whence tang. g=

Y

Let R be the intensity of the repulsive action exercised between the poles of the same name, A and A'; R' that of the repulsive or attractive action exercised between B and A'. The first, making with the axis of the positive x an angle of 180°-i, will have, according to the axes of x and y, components R cos. (180°—i), or h x and R, remarking

-R cos. i, and R sin. i; or, finally, -R

y

that

sin. i. The components of R', acting like the preceding

on the side of the positive co-ordinates, will be found in like

h + x

manner, and will be Fr1‡ and FR by taking, as we have

FR''

hitherto done, the upper or the lower signs, according as в is of the contrary or of the same name with A, and according as R' represents an attraction or a repulsion. The components of the total action exercised upon a' will therefore be,

By equalising this value with that which we had found,

tang.g=

y(r's p3)
y'3 ( h − x ) ± 13 ( h + x) *

whence results the relation

(Rr'FR′r) (r3(h—x)±r3 (h+x)) — (Rr'(h−x)+R'′r(h+x))

(r'3Fr3)=0.

By developing the calculations and separating the terms that contain h-x and h+x, it becomes

(FR′rr'3±Rr3 r') (h−x)±(+Rr′r3 — R′r r′3) (h+x)=0;

or by reducing and dividing by 2hrr',

Rr2=R'r'2.

This equation, in which R and R' represent functions of r and r respectively, occurs in all the points of each magnetic curve; and as one may be passed through every point of space, it occurs for any point whatever, r and r' designating distances to A and B. We may therefore maker vary arbitrarily by leaving to a constant value; representing by H the independent value of r then

taken by R'2, we shall have, whatever r may be, Rr2=H; whence,

Consequently the intensity R of the mutual action of the poles A, A', varies in inverse ratio with the square of their distance r.

NOTE E. (p. 217.)

Note relative to the Law of the Action of a Point of an Electric Current upon the Magnetised Needle.

EXPERIENCE demonstrates that the intensity of the action, exercised by an indefinite rectilinear current upon a magnetised needle, is in inverse ratio to the distance of the needle from the current. We have said the consequence of this experimental law is, that the elementary action of a simple point, or of a section of the current, is in inverse ratio, not to the simple distance, but to the square of the distance.

Let м N be the indefinite current; A, the centre of oscillation, that is to say, the middle of the magnetised needle, the point upon

B

M

N

which we may suppose the action of the
current as being concentrated; let a B, or c,
be the shortest distance from the needle to
the current; finally, let a c be the distance
from any point of the current to the needle.
The action of the point c, by supposing the
law of the inverse of the square, will be
hds
AC 2

expressed by

;

h being a constant, depending on the mutual force of the current and the needle; and ds the rectilinear element of the current whose action is being investigated. But AC2=CB2 + BA2=s2 + c2, s being the distance CB from any point c to the fixed point B; the quantity s is a variable, because the form of the expression which gives the value of the action is the same, whatever be the point of the current that is

Fig. 5.

taken. We have, therefore, only to integrate this expression

hds

s2+ c2

to obtain the value of the total action of the

indefinite current MN; since we suppose that it extends indefinitely, setting out from в in both directions.

μπ

expression becomes ; a quantity in which h and are con

с

stant, and in which c only, namely the distance from the needle to the current, may vary. The action of an indefinite rectilinear current upon a magnetised needle is found, therefore, to be, as we have said, the inverse of the simple distance from this current to the needle, when we suppose that the action of a simple element of the current is in inverse ratio to the square of this distance.

Calculation likewise demonstrates that if the current is angular instead of rectilinear, and we admit that the action exercised by an element of this current upon a magnetised particle varies, not only in inverse ratio with the square of the distance, but also proportionately to the sine of the angle made with the direction of the current by the line joining the centres of the element and of the particle, we arrive at the result that the total action of the current is not only, as we have just seen, the inverse of the simple distance, but also proportional to the tangent of half the angle formed by the current that has become indefinite. This second result of calculation has been verified, like the former, by an experiment of M. Biot's. We shall not give here this latter calculation, in which the same course is pursued as for the former.

NOTE F. (p. 246.)

Note relative to the Calculation of the mutual Action of two Electric Currents.

THE very great development that I have given to this note is due to my desire to make known, in a tolerably complete manner, by endeavouring to place within the reach of the generality of readers the admirable mathematical theory upon which Ampère has based the explanation of all electro-dynamic phenomena; a theory so perfect that, by its means, the celebrated philosopher to whom it is due was able so to predict, even in their minutest details, all