It is here supposed that lies between 0 and 7; if it lies between π and 2π, we have the same sum,

To find the sum for greater values of 7, it is to be remembered that it is periodic as regards 2.

From this it appears, that at every moment a point exists in the wire in which the intensity of the current suffers a sudden change or break. This point, at the time t=0, lies at the end of the wire, but moves from this with the velocity towards

с

the commencement, after reaching which it returns with the same velocity towards the end; turns here again, and thus travels perpetually to and fro over the length of the wire. In each of the two portions into which the wire is at each moment divided by this point, the same intensity exists everywhere at that moment; so that if s and i be regarded as rectangular coordinates of a point, a line is described of the form of fig. 1. The intensity before the point at which the break occurs, considered without regard to its sign, is always the smaller, that behind the point the greater, the words before and behind being used with refer

Fig. 1.

ence to the direction in which the point moves. The figure 1

is therefore true only

for a moment in which

the point moves from the end towards the commencement of the wire. The figure 2 re

Fig. 2.

2

fers to a moment in which the opposite takes place.

The mag

Phil. Mag. S. 4. Vol. 13. No. 88. June 1857.

2 F

or if we denote by J the value to which i approximates as the

time is increased, that is, the value of

K

This quantity has its greatest value when t=0; but this, in accordance with an assumption already made, is also infinitely small in comparison to J. The expression for the magnitude of the break may be more shortly written, when the time is introduced required by the point at which it takes place, or the time required by an electric wave to move through the length of the wire. Denoting this time by T, that is, setting

As the time increases, the magnitude of the break diminishes, but so slowly that during the time T only an infinitely small diminution takes place.

To obtain a complete view of the process, it is now only necessary to examine the alterations of the strength of the current at the commencement of the wire. Let this, that is to say, the value of i for s=0, be i。; then making use of the symbols J and T, we find

Setting for the sum its value, and remembering that

i。=J(1− e−2ht)+J2he-ht (2pT−t),

where p denotes the whole number for which

t-2pT
T

is a proper fraction, positive or negative. p may also be defined as the greatest integer which is contained in the fraction

t+T
2T

For values of t, for which the number p is not very great, the expression for io is capable of a considerable simplification. For such the quantity ht is infinitely small; and by neglecting members of higher orders, the equation for i。 may be thus written: i=J.2ht+J2h(2pT−t),

that is,

i=pJ4hT.

This expression shows that the intensity at the commencement of the wire is 0 up to the time when t=T; here and at the times t=3T, t=5T, &c., it alters itself by jumps; and moreover the jump is twice as great as at other points of the wire. During the intervening times the intensity is constant.

In a similar manner the expression for e may be discussed. We have

if lies between π and 2π, we have the same sum,

T

The second fact follows from the first, when it is considered that the sum has the same value for 7 and for 2π-T. For greater values of 7, the value of the sum is found when we remember that it is periodic with reference to 2π.

From this it follows that at each moment at some one point of the wire, e also suffers a break. This point always coincides with that in which the break for i takes place. e is always greater on the side of this point on which the end of the wire lies, and smaller on the side of the commencement. The magnitude of the break is

or, denoting by E the constant value of e at the end of the wire,

= Ee-ht.

At that side of the break on which the commencement of the

wire lies, we have

e=E.†(1—e−1);

and on the side towards the end,

=E{ † (1− e−k) +e=ht}.

If e and s be made the rectangular coordinates of a point, then for a certain value of t we obtain a line of the form shown in fig. 3; when t does not exceed a moderate multiple of T. the line has the form shown in fig. 4; the more t increases, the

more nearly does the

Fig. 4.

Fig. 3.

figure approximate to the straight line, fig. 5.

Fig. 5.

e

LV. On Parallel Lines. By J. P. NICHOL, Esq. To the Editors of the Philosophical Magazine and Journal.

Observatory, Glasgow,
May 6, 1857.

GENTLEMEN, HE frequent appearance of late in your publication, of memoirs concerning the doctrine of parallel lines, assuring me that the subject continues to possess an interest, I have ventured to request insertion for the enclosed article, which has just been printed in a 'Cyclopædia of the Physical Sciences,' edited by myself. It contains views not touched by your correspondents. I shall not say that those views do, in my own opinion, exclude the applicability of other modes of solving the difficulty, or that they are in themselves free from objection. Nevertheless, I think that principles are indicated there, which must command a prominent place in any successful attempt to regenerate our elementary geometry. It is certainly out of rule

to ask you to accept an extract from a published work: but, in the present case, the volume is only just published; and as it is of a miscellaneous nature, the brief article of which a copy is now sent, may easily escape the notice of those of your correspondents who take a critical interest in the very curious subject under discussion.

"The characteristic of two lines in the same plane, to which the name of parallel is given in geometry, is simply this,—that although produced ever so far either way they will never meet. The theory of these lines continues a stain on our elementary science. It is easy to prove, that if certain conditions are fulfilled when two lines cut a transverse line, these two will never meet, or must be parallel; but to establish the converse,-to prove, viz. that if the two lines are parallel, these same conditions are lawfully predicable,-has hitherto defied the logic of all geometers-a fact certainly most remarkable in this purely deductive science: nevertheless, its causes do not appear remote. The existence of such a defect unquestionably argues some oversight in the list of geometrical axioms,-the oversight of the nature of some of our primal perceptions regarding magnitude: but it does not follow that the missing axiom has immediate relation to parallels, or that it ought to enable us to resolve directly the specific proposition at which the acknowledged difficulty first appears. On the very contrary, it may be asserted with abundant confidence, that nothing but failure could attend the effort -originated by Euclid, and since his time all but universally followed-to supply the deficiency by new postulates or axioms regarding parallel lines. To prove that under certain conditions two lines will never meet, or what is the same thing, that no triangle can be formed in such circumstances, involves no conception with which we cannot readily cope; but to deduce the properties of two lines postulated as parallels, involves a direct dealing with the positive idea of infinity-a task utterly beyond the reach of our faculties. Whether we have a positive idea of the Infinite, is a question concerning which the profoundest metaphysicians have differed and continue to differ; but, consideration of the origin and formation of language, suffices of itself to leave no doubt, that we cannot speak of Infinity otherwise than as a negation, and therefore that no positive axiom can be laid down respecting it,