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seems to have been independent of Legendre's; and I had myself arrived at the same result before I had seen what these mathematicians had written on the subject. See pages ix and 427 of my History of the Calculus of Variations.'

The problem as discussed by Professor Challis is free from this condition; it may be enunciated thus:-required to connect the ends of a fixed straight line by a curve of given length so that the area bounded by the curve and the straight line may be a maximum. The result is well known, namely that the curve must be an arc of a circle. The given length must, of course, be greater than the length of the fixed straight line; Professor Challis by a misprint has less instead of greater. The problem as thus enunciated is one of the oldest and most familiar in the subject; and I believe there has never been any doubt or difficulty as to the result, which may be obtained by various unexceptionable methods; I have indicated one of these methods at page 69 of my 'History.'

I do not accept the results at which Professor Challis arrives with respect to the other two problems he discusses; but I have not leisure to enter into details. I have, I believe, fully solved these problems in an Essay which will be published in the course of the present month.

St. John's College, Cambridge,
November 6, 1871.

I. TODHUNTER.

LVII. On a Correction sometimes required in Curves professing to represent the Connexion between two Physical Magnitudes. By the Hon. J. W. STRUTT, M.A.

THE nature of the correction which is the subject of the

present paper, and of not infrequent application in experimental inquiry, will be best understood from an example, as it is a little difficult to state with full generality. Suppose that our object is to determine the distribution of heat in the spectrum of the sun or any other source of light. A line thermopile would be placed in the path of the light, and the deflection of the galvanometer noted for a series of positions. But the observations obtained in this way are not sharp-that is, they do not correspond to definite values of the wave-length or refrac tive index. În the first place, the spectrum cannot be absolutely pure; at each point there is a certain admixture of neighbouring rays. Further, even if the spectrum were pure, it would still be impossible to operate with a mathematical line of it; so that the result, instead of belonging to a simple definite value of the

* Communicated by the Author.

independent variable, is really a kind of average corresponding to values grouped together in a small cluster.

For the sake of simplicity, let us suppose that the spectrum is originally pure, and that the true curve giving the relations between the two quantities is PQR. Also let M N be the range over which the independent variable changes in each observation -in our case the width of the thermopile. Then the observed curve is to be found from the true by taking m, the middle point of M N, and erecting an ordinate pm, such that

pm. MN area of curve PQN M.

=

The locus of p will give the curve expressing the result of the observations. It remains to find a convenient method of passing from the one curve to the other.

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In the figure PRQ represents the true curve, MN the range as before; M mm N=h; p is the point on the observed curve found in the manner described; Om=xo, Rm=y。 pm=y. Now

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(A) and (B) give the analytical solution of the problem; but for practical purposes the following interpretation is important:

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In passing from the observed to the true curve, the curvature is everywhere to be increased instead of diminished, and

pm=Rm+4R S.

It may be remarked that while it is always possible to pass accurately from the true curve to that derived from it with any prescribed range, the inverse problem is not determinate unless it be understood that the range is small, so that its fourth power may be neglected. The practical utility of the solution obtained is scarcely affected by this consideration; indeed it is only when the curvature of the curve is considerable that the correction itself is of much importance.

It often happens that the connexion between the two curves is not so simple, at least at first sight. Suppose, for example, that in the case taken as an illustration the spectrum is impure from the sensible width (2x) of the image of the slit. The observed curve is then connected with true by a double integration,

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if the term in h2x2 may be neglected. Thus

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The rules remain just as before, except that instead of h we

now have √2+2. Similarly, when the want of sharpness is due to more than two causes, we must replace h by {Σh2}}. When, as often happens, the product of the quantities he... is to be considered as given, the experiments are best arranged so as to make the independent quantities equal; for then the agreement between the two curves is the closest.

The practical rule to which we are led by the considerations explained in this paper is therefore as follows:

Construct the curve representing the immediate results of the observations in the ordinary way. Let Rm be any ordinate. Draw

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parallels PM, QN at distances equal to h, or {Σh2}1, and join PQ cutting Rm in S. The point p on the true curve corresponding to the abscissa Om is to be found by taking p R equal to one third of R S, and so that p and S lie on opposite sides of R.

LVIII. On a Method of Measuring the Lateral Diffusion of a Current in a Conductor by means of Equipotential Lines; being a Note on a Paper by Dr. Macaluso "On the Transmission of Electricity in Liquids," in the Philosophical Magazine for November 1871. By J. E. H. GORDON*.

IN

N the abstract of Dr. Macaluso's paper given in the Philosophical Magazine for November 1871, p. 389, it is stated that "when an electric current travels through a liquid whose cross section is much greater than the surface of the electrodes, it tends to diffuse itself laterally." About two years ago I made some experiments in the physical laboratory of King's College, on the lateral diffusion of the current in conductors. Instead of liquids, however, I used sheets of tinfoil of various shapes.

I was not only able to experimentally detect the lateral diffusion, but also to arrange a method by which its direction and amount may be measured. The outline of the method was suggested to me by a friend. I do not know whether it is new; but as I have * Communicated by the Author.

not seen any description of it, and as many details were arranged by me, I think it may be worth while to publish a short account of it.

The law of the diffusion of the current in a solid conductor is of course perfectly well known. In this paper I merely propose to describe a method of experimentally verifying this known law, and to suggest a modification of it by means of which the (apparently) unknown law for the diffusion of the current in a liquid may probably be determined.

The object of the method was, on a surface of tinfoil through which a current was flowing, to determine a number of groups of points of equal tension, and to draw equipotential lines through them. From these lines the positions of the currents can be easily deduced.

My plan was as follows:-A sheet of tinfoil of any desired shape was pasted on to a hard mahogany board. At the two points chosen for the electrodes holes were drilled, through which screws were passed so that their heads (about 7 millims. in diameter) rested on the tinfoil; their stems passed through the board into binding-screws on the underside. A little mercury was placed under the heads to make better contact. Wires from a battery passed through a contact-breaker to these screws. Two cells of Grove's battery were at first used; but the power was afterwards reduced to one, owing to the great heating-effect of two cells.

The contact-breaker was arranged so that two circuits could be broken or made simultaneously, namely the battery and galvanometer circuits. A current is produced simply by difference of tension. Therefore, if two poles of a sufficiently sensitive galvanometer were placed in such a position on the tinfoil that there was no deflection when the current passed, these two points were in the same equipotential line.

The beautiful reflecting galvanometer belonging to King's College was used, and was found to be so delicate that, after a little practice, a deviation of less than 1 millimetre from the right position on the tinfoil could be easily detected on the galva

nometer.

The same plan can be used for determining the lines in liquid conductors by insulating the wires up to their points, and having each galvanometer-terminal fixed on a little stage to slide on the top of the trough, which must be divided so that from the position of the stage the horizontal coordinates of the point of the wire can be determined, while the wire slides through the stage and is divided to give the vertical element.

Below is an engraving of half one of the sheets of tinfoil with the lines traced on it. The sheet is divided by a line at right angles to, and bisecting, the line joining the battery-poles. The

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