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so that, in effect, we may in the above values of A, A,, &c. put c=1, and c1=c2=&c. =0; i. e. we shall have

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as to which it is to be observed that no constants are to be introduced in the integrations, all such having been accounted for. Take the case where R1, and S and T are constant, and we get

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And the substitution of this value in (8) gives us

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40 On Linear Partial Differential Equations of the Second Order.

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{k+h(h−1)} {k+(h−1)(h—2)—l} xh-2

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1.2

Similarly we shall arrive at

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Ag:

{k+h(h−1 ) } {k+(h−1)(h−2)—l} {k+(h−1)(h−2)−21}_xh-3

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A,=

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{k+h(h−1} {k+ (h−1) (h−2) −l} ... {k+ (h−r+2) (h−r+1) — (r−1)}};

xh-r 1.2...r

If h is fractional or negative, we shall, on the particular assumptions above introduced, always have an integral of the assumed form, the number of terms being finite or infinite according to circumstances, though as to the practical value of the integral so obtained in the latter case I am not prepared to express an opinion.

The condition to be satisfied in order that A, may vanish and that the expression for z may have a finite number of terms when h is not a positive integer, is, that we have

0=k+(h−r+2) (h−r+1) — (r−1)l,

the only conditions limiting the quantities h, k, l, r being that
they are all constant, and that r is, and h is not, a positive
integer.

When h is a positive integer and r=h, we shall have A, con-
stant; whence it follows that upon this supposition the series
will always terminate when U=0.

The well-known equations

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are readily solvable by the foregoing method.

It remains to be remarked with respect to the first term of the series for 2, that since (3) is of precisely the same form as (1), any value assigned to A must be different from any derivable from the general expression for 2-as, for instance, a solution obtained upon a particular hypothesis not necessarily implied by

On a Simple Method of Constructing high Electrical Resistance. 41 the terms of the problem submitted to us-for example, on the assumption that one of the partial differential coefficients p, q, r, s, t vanishes.

6 New Square, Lincoln's Inn, June 20, 1870.

VI. On a Simple Method of Constructing high Electrical Resistance. By SAMUEL E. PHILLIPS, Jun.*

AT present resistance-coils are mostly made with German

silver wire; and a set of coils equalling 10,000 B.A. units forms a box of convenient size. But in electrical research resistances of several million units are often useful; and to produce such with wire in the ordinary way would be both expensive and cumbrous.

Latterly Mr. Hockin has used selenium for this purpose. Fine glass tubes with a platinum wire blown in at each end and filled with different fluids, according to the resistance required, have also been largely used; but these latter necessarily give very variable results, owing to polarization and electrolysis; and the former, I believe, are somewhat difficult to construct.

Requiring a high resistance for some experiments, I made one as follows:-Upon a strip of vulcanite, 6 inches long by 1 inch wide, I ruled several pencil lines with an ordinary H.B. pencil in such a manner as to produce a continuous line about of an inch wide and 4 inches long. At the extremities of the line I rubbed the pencil plentifully over a space as large as a sixpenny piece, upon which I firmly screwed two binding-screws by means of "nuts" underneath, and, gently dusting off all superfluous plumbago, varnished the whole with several coats of pure shellac varnish.

The above arrangement gave me a resistance of slightly over two million B.A. units. It was constructed three months ago, and up to the present time the resistance has remained very constant. I have tested it repeatedly with 100, 200, and 300 cells, and have always obtained the same result within very small limits. It is also beautifully steady with prolonged battery

contact.

Mr. G. Preece has kindly tested a resistance made in the above manner for me, and finds it very constant, only getting an alteration of about 0.5 per cent. for 5° F. I have mounted a vulcanite slab with twenty binding-screws, giving a wide range of varied resistances by combination or otherwise, and hope shortly to make some experiments with the view of determining the ratio of its alteration by difference of temperature.

* Communicated by the Author.

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VII. Researches on the Electrical Discharge.

By Professor von Bezold*.

N the course of the further investigation on the connexion I have recently described† as existing between the mode of discharge and the character of the dust-figures thereby produced, I strongly felt the necessity of producing the phenomena in question by a more simple apparatus than Ruhmkorff's.

The first experiments with charged Leyden jars, as well as with the ordinary electrical machine without condensing-arrangements, soon showed that with these means only simple figures (that is to say, discharges) could be obtained.

The observation of the spark is sufficient to prove that the discharge which, with a good conducting circuit only interrupted by a break, is alternating, is changed into a simple one by inserting a test-plate; for while in the first case the spark is brilliantly luminous, it appears in the second only as a narrow purple line with a bright point towards the positive electrode.

In order, therefore, to obtain alternating discharges even when the test-plate was interposed, no way was left but the use of a suitable branch or return conductor.

If this conductor (which goes to earth) is continuous (that is, nowhere interrupted by a break), it is to be expected that the discharge of the conductor through which the electricity is led to the plate will ensue directly after the charge,—that is, that in this conductor one or more alternations of electricity will take place.

In experiments made with such return conductors, various entirely new and surprising facts were observed which seem suited to serve as starting-points for new inquiries.

But, before I begin the description of these new facts, I must first mention a simple experiment, which indeed teaches nothing essentially new, but yet contributes materially to the understanding of the following.

If the otherwise insulated coating of the test-plate is placed in conducting communication with the source of electricity, while the needle which at other times serves as conductor is connected with the earth, a positive discharge upon the glass surface produces a negative figure, and conversely.

If the coating be perfectly insulated, while two conductors (A * Communicated by the Author, having been read before the Bavarian Royal Academy of Sciences, February 5, 1870.

f Phil. Mag. S. 4. vol. xxxix. p. 392.

By test-plate (Probeplatte) I shall in the sequel mean the plate, coated on one side, on which the figures are formed.

and B) are placed upon the upper uncoated surface, one of which is connected with the source of electricity Q (fig. 1), and the other by a wire E with the earth, at each discharge a positive and a negative figure will be simultaneously formed.

These experiments teach that a positive (negative) figure is obtained when either positive (negative) electricity

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is added, or negative (positive) taken away.

Fig. 1.

F

B

§1. This being premised, the above-mentioned experiments shall now be described. One of the first was made according to the following scheme (fig. 2). From the positive conductor of

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an electrical machine a wire was led to one knob of a sparkmicrometer F. From the other knob two wires conducted, one (E) directly to the earth, the other (D) to the conductor A. The lower coating of the plate was also connected with the earth by means of the wire E'. In my opinion, two kinds of results were by this means to be expected. For it was conceivable that either no figure at all would be formed upon the plate, and the whole of the electricity would be immediately conveyed to the earth by the good conducting-wire, or that at most a small part would reach the plate and then again pass back through E to the earth. I therefore expected either no figure at all, or a small positive compound figure—that is, a yellow star with a red spot. The result was nevertheless the exact opposite. A figure was formed; it was not positive, however, but negative, a red irregular jagged ring with a yellow radiating centre.

Hence the discharge had not only not divided at the two branches, but the electricity, flowing to the earth in the shortest way through E, took with it electricity of the same kind from the branch A E'.

Not only the very surprising nature of the experiment, but

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