of the dimensions of the Atlantic cable, either actually submerged or placed in perfectly similar inductive circumstances.

II. Method for telegraphing through submarine or subterranean lines of not more than 500 miles length.

The plan which I have proposed to describe for rapid signalling through shorter wires, has one characteristic in common with the plan I have already suggested for the Atlantic telegraph; namely, that of using different strengths of current for different signals.

But in lines of less than 500 miles, condensed pulses, such as have been described, may be made to follow one another more rapidly than to admit of being read off by an observer watching the image of a scale in a suspended mirror; and a new plan of receiving and recording the indications becomes necessary.

Of various plans which I have considered, the following seems most likely to prove convenient in practice.

Several small steel magnets (perhaps each about half an inch long) are suspended horizontally by fine threads or wires at different positions in the neighbourhood of a coil of which one end is connected with the line wire and the other with the earth. Each of these magnets is held in a position deflected from the magnetic meridian by two stops on which its ends press; and two other small stops of platinum wire are arranged to prevent it from turning through more than a very small angle when actuated by any deflecting force making it leave the first position. When a current passing through the coil produces this effect on any one of these magnets, it immediately strikes the last-mentioned stops, and so completes a circuit through a local battery and makes a mark on prepared electro-chemical paper. For each suspended magnet there is a separate style, but of course one battery is sufficient for the whole printing process. One set of the different suspended magnets are so adjusted, that a current in one direction of any strength falling short of a certain limit makes only one of them move; that a current in the same direction, of strength exceeding this limit but falling short of another limit, moves another also of the suspended magnets; and so on for a succession of different limits of strength of current in one direction, The remaining set of suspended magnets are adjusted to move with different strengths of current in the other direction through the coil. Without experience it is impossible to say how many gradations of strength could be conveniently arranged to be thus distinguished unmistakeably. I have no doubt, however, that very moderate applications of electric resources would give at least three different strengths of current in each direction, which could with ease and certainty be distinguished from one another by the test which the suspended magnets afford. Thus, a signal of six varieties -one letter of an alphabet of six-could be recorded by almost instantaneous movements of six suspended magnets, making one, two or three marks by one set of three styles, or one, two or three marks by another set of three styles, placed all six beside one another, pressing on a slip of electro-chemical paper drawn by clockwork, as in the Morse instrument.

In subterranean or submarine lines of less than 100 miles length, it would be easy, by means of simple battery applications, followed by connexions with the earth, or by means of simple electro-magnetic impulses at one end of the wire, to give ten or twelve of such signals per second without any confusion of utterance at the other end. The confusion of utterance which would be experienced in working thus through longer lines would be easily done away with, in any length up to 500 miles, by following up each battery application with a reverse application for a shorter time, or by following up each electromagnetic impulse by a weaker reverse impulse, so as approximately to fulfil the condition (described in my former communication), of reducing the subsidence of the electrification in the wire to the double harmonic form. It would, I believe, be readily practicable to send distinctly five or six such signals per second (each a distinct letter of an alphabet of six) through a wire of 500 miles length in a submarine cable of ordinary dimensions. To perform the electrical operations required for sending a message on this system, mechanism might be had recourse to, and, by the use of perforated slips, as in Bain's and other systems, it would be easy to work from twelve to twenty of the six-fold varied signals per second through lines of less than 100 miles length. Operating by the hand is, however, I believe, generally preferred for ordinary telegraphing; and no such speed as the last-mentioned could be attained even by a skilful operator working with both hands. Six distinct letters or signs of an alphabet of thirty, could, however, I believe, be delivered per second by the two hands working on a key-board with twelve keys (perhaps like those of a pianoforte), provided the keys are so arranged as to fulfil the following conditions:-

(1) That by simply striking once any one of a first set of six of the keys, an electric operation of one or other of the six varieties shall be made twice, the second time commencing at a definite interval (perhaps th of a second) later than the first.

(2) That by striking one or other of the remaining six keys at the same time, or very nearly at the same time, as one of the first set, the second operation of the double electric signal will be that corresponding to the key of the second set which is struck, instead of being a mere repetition of the operation corresponding to the key of the first set.

It would certainly be easy to make a key-board to fulfil these conditions with the aid of some clockwork power. Then by arranging the thirty-six permutations and doubles of the six simple signals to represent an alphabet of thirty-six letters and signs, an experienced operator would have to direct his mind to only six different letters per second, while executing them by six double operations with his fingers. That it would be possible to work by hand at this rate there can be no doubt, when we consider the marvels of rapid execution so commonly attained by practice on the pianoforte; and it appears not improbable that in regular telegraphic work, practised operators of ordinary skill could perform from four to six letters with ease per second, or from forty to sixty words per minute, on lines of not more than 100 miles length. The six signals per second, which, according

to the preceding estimate, could be distinctly conveyed by a submerged wire of 500 miles in length, could of course be easily performed by the hand, with the aid of a key-board and clockwork power adapted to make the double operations for giving rapid subsidence of electricity in the wire when any one key is touched, and to let the different strengths of current, in one direction or the other, be produced by the different keys. Thus without a condensed code, thirty words per minute could be telegraphed through subterranean or submarine lines of 500 miles; and from thirty to fifty or sixty words per minute through such lines, of lengths of from 500 miles to 100 miles.

The rate of from fifty to sixty words per minute could be attained through almost any length of air line, were it not for the defects of insulation to which such lines are exposed. If the imperfection of the insulation remained constant, or only varied slowly from day to day with the humidity of the atmosphere, the method I have indicated might probably, with suitable adjustments, be made successful; and I think it possible that it may be found to answer for air lines of hundreds of miles' length. But in a short air line, the strengths of the currents received, at one extremity, from graduated operations performed at the other, might suddenly, in the middle of a message, become so much changed as to throw all the indications into confusion, in consequence of a shower of rain, or a trickling of water along a spider's web.

"On the Equation of Laplace's Functions," &c. By W. F. Donkin, M.A., F.R.S., F.R.A.S., Savilian Professor of Astronomy, Oxford.

d'u

d'u d2u The equation + + =0, when transformed by putting dx2 dy2 dz2

x=r sin 0 cos p, y=r sin 0 sin 4, z=r cos 0, may be written in the form

n?

and if u=u+ur +μ‚2 + ... + up2+..., we find on substituting this value in (1), and equating to zero the coefficient of ", that u, satisfies the equation

d 2 d 2

{ (sin e ~)2+()*+n(n+1) (sin 0)2

} ~

Un=0,

commonly called the equation of Laplace's functions. If we put sin 0 +n cos 0=,, then the equation (2) may be written

and the operation w, possesses the following property, namely

@_nwn+n2=wn−1 w—(n−1), + (n − 1)2;

hence it is easily shown, that in general the complete solution of (2) is

Phil. Mag. S. 4. Vol. 14, No. 90, July 1857.

F

and the operation ww-1... ww, is easily seen to be equivalent to

d

10)".

sin 0 (sin 0)~" (sin 0

(This result is compared with that obtained in a different way by Professor Boole (Cambridge and Dublin Journal, vol. i. p. 18), to which it bears a general resemblance, but the author has not succeeded at present in reducing the one form to the other.)

In the case in which un does not contain ø, we have

0 =C1+C, log tan.

The general expression for a "Laplace's coefficient" of the nth order,

d

not containing ø, is therefore (sin 0)-" (sin e sine)".

do

e)". C; and if this be called when C=1, the development of (1-2rcos 0+r3) −† is

+ +Un

1.2

доп 1.2 n ...

+...;

and it is shown that the coefficient of

доб
1.2 n
...

in the development

i+1

of (1-2r cos 0+) 2 is

d de

(sin 9) -"-t (sin 0 sin 0)"(sin 8)'.

With respect to the development of

(1—2r(cos e cos 0' + sin@ sin@'coso)+r2)−*,

it is shown that the coefficient of " cos ip may be put in either of the

of which the complete integral may be expressed in the form

at least in the case in which i is an integer not greater than n, for which case this form is here demonstrated.)

If it be assumed that the solution of (2), obtained on the supposition that n is an integer, may be extended to the case in which n is a general symbol, it follows that the solution of (1) will be obtained This would give

from it by changing n into r

d

-P dr

d

dr

d

u= (sin 9)TM*** (sin 0% sin

+F(1, e-0√=1 tang)}.

which is easily shown to be equivalent to

8

d

+F

d

u=( sin 8 sine, etan 2)+P(o sinine, e-tang),

where pr(sin 0), but p is to be treated as a constant till after all operations.

This expression is shown to give known particular integrals, such as (1-2r cos 0+), and

d

i

2 cos ip.

It appears probable, therefore, that the generalization of the result obtained for the limited value of n is legitimate; but the author does not profess to demonstrate this conclusion, believing that the principle of the "permanence of equivalent forms" is not at present established in such a sense as to amount to a demonstration.

"A Memoir on Curves of the Third Order." By Arthur Cayley, Esq., F.R.S.

A curve of the third order, or cubic curve, is the locus represented by an equation such as U=(*(x, y, z)3=0; and it appears by my "Third Memoir on Quantics," that it is proper to consider, in connexion with the curve of the third order, U=0, and its Hessian HU=0 (which is also a curve of the third order), two curves of the third class, viz. the curves represented by the equations PU=0 and QU=0. These equations, I say, represent curves of the third class; in fact, PU and QU are contravariants of U, and therefore, when the variables x, y, z of U are considered as point coordinates, the variables E, n, of PU, QU must be considered as line coordinates, and the curves will be curves of the third class. I propose (in analogy with the form of the word Hessian) to call the two curves in question the Pippian and Quippian respectively. A geometrical definition of the Pippian was readily found; the curve is in fact Steiner's curve R, mentioned in the memoir "Allgemeine

F 2