The number given by Grover is log q=14035322; the value for z/A is 0.38353439, while Grover gives for it 0.3835341, agreeing to within a ten-millionth part.

For the convenience of practical calculation, the following tables of Ar and r are given for different values of q and 91 respectively, calculated for me by Mr. Shobei Shimizu Ar in the first table are residuals

614-49-7934 69+....-2301 × 106,15

in (II.), while r in the second table refers to those calculated according to (II'.).

;

From the table, it will be noticed that the residuals apparently cover a wide range in the calculation with q; this is at once evident from the fact that the range covered by the q-series is very great compared with that in which the 91-series are applicable. This is in no way an impediment in practical calculations, as the formula in terms of q is simpler than that in q1; especially in numerical evaluation of q from known values of r, the formula (II.) is characterized by the great facility with which all the terms can be calculated and the required approximations brought to test. As all the rest of the calculation depends on the value of q,

20 Maximum Force between Two Coaxial Circular Currents.

when once this is accurately known, the distance between the coils and the force exerted at that distance can be found by the formulæ (1) and (6).

As regards the utility of the solution of the present problem, it would be unnecessary to spend words on its bearing in the construction of the current-balance for the absolute measurement of electric current. The numerical data calculated by Grover are of great value in researches of this kind. The solution which I have here given may be of service in the direct calculation when the dimensions of the coils are given. How far the accuracy of the instrument can be relied upon is of great interest to me, as I believe that the instrument can be used for a purpose totally different from the usual measurement of the electric current, and which seems not yet to have been well noticed.

The most exact method of measuring relative values of gravity is that of comparing the periods of invariable pendulums at the place of observation with those at the standard station. The great inconvenience and difficulty accompanying the method of observation lie in the extremely accurate measurement of time; the rate of the clock must be known to 1/60th part of a second per day, if the period is to be exact to one part in five million. For this we have to take a transit instrument of fairly large aperture, and when obstructed by bad weather we have to wait for days. In addition to this, the occasional change of the clock-rate necessitates the unintermittent continuation of observation which imposes a great burden on the observer. This tedious and unwelcome obstruction to the usual method of gravity determination may, to a great extent, be overcome by using a current-balance instead of invariable pendulums. The strength of the current is to be evaluated by means of known resistance of the circuit and the terminal potential difference, for which the electromotive force of the cadmium cell must be relied upon. It is a question if we can bring the constancy of the cadmium cells and of the coils to the same order as that of the pendulum and clock. The weight counterbalancing the attraction of the coils is an immediate measure of the force of gravity at the place of observation. For this purpose it is perhaps necessary to design an instrument anew in a transportable form, and construct the coils such that the attraction is of sufficient amount to give the desired accuracy. It must however be well noticed that the method of current-balance is not free from objections, as the current is liable to fluctuations and the coils are heated in course of measurement and give rise to convection current; moreover, the correction to be applied to such disturbances

is difficult to calculate and almost impossible to estimate exactly. There may be other methods of dispensing with astronomical observations in gravity measurements, but I believe that the method of current-balance is one of the most accurate that can be easily brought into practice without sacrificing to any great extent the degree of precision usually attainable with invariable pendulums.

In the theory of atomic constitution, it is generally assumed that there are rings of electrons in rapid rotation: these are no doubt equivalent to currents and must exert mutual influence upon each other. When such atoms combine into a molecule and are in such a position that the planes of the rings are parallel to each other, then the position of maximum force between two circular rings here discussed will be of some significance in the atomic configuration of molecules.

III. On the Nodal-Slide Method of Focometry. By J. A. TOMKINS, A.R.C.S. (Lond.), Lecturer in Physics, Technical College, Bradford*.

PROFES

ROF. A. ANDERSON has recently described (Phil. Mag. Jan. 1917, p. 157) an elegant method of determining the focal length, and other constants, of a lens system based upon a general theorem of which the ordinary nodalslide method is but a particular application. This theorem is that for any lens system there is always one, and only one, point on the optic axis such that a small rotation of the system about a perpendicular axis through it will cause no lateral displacement of the image of an object in a given position. This point was shown by Prof. Anderson to divide the distance between the object and image, and also that between the nodal points, externally in a ratio equal to the value of the magnification.

A numerical example of the determination of the focal length of a diverging combination by this method was given, but it was pointed out by Mr. R. E. Baynes (Phil. Mag. April 1917, p. 357) that these data yielded very different results for the focal length and the positions of the nodal points when calculated in different ways. These discrepancies were explained by Prof. Anderson (Phil. Mag. July 1917, p. 76), who discussed the effect of various errors and showed that, while the method gave quite satisfactory values for the focal length, it failed to do so for the distance between

* Communicated by the Author.

the nodal points, because small errors are multiplied by the numerical value of d, the distance through which the combination is moved, which may be large. He also described another method which gave satisfactory values for this distance. In a third paper (Phil. Mag. Sept. 1917, p. 174) he gave some further properties of this point, which he terms the nul point. There are, however, two possible sources of error mentioned by Prof. Anderson, viz. (1) want of precision in determining whether there is any displacement of the image, and (2) error in determining its position, which seem to call for further consideration.

With reference to the first it is to be noted that in the ordinary nodal-slide method there is one, and only one, possible axis of rotation of the lens system, viz. that passing through the second nodal point, whereas in the general method described by Prof. Anderson there is an infinite, or doubly infinite, number of possible axes. The object of this communication is to investigate the best position, if any, for the nul point, and to compare the results with those obtained by the ordinary nodal-slide method.

For the purpose of observing the displacement of the image the best position will be that for which a given small displacement of the axis from the nul position will, for a given small rotation of the lens system, produce the greatest displacement of the image.

To determine this it is necessary first to find an expression for the displacement of the image due to a small rotation about any axis.

Fig. 1.

Fig. 1 shows the displacement produced by a convergent combination in the general case in which the first and last media are different, and in which, therefore, the principal and nodal points are not coincident. P,Q, and P,Q, are the object and image respectively, H1 and H2 the principal

2

2

2

points, N1 and N, the nodal points, F, and F, the principal foci. Suppose the system to be rotated about O through a small angle so that the principal axis moves into the position indicated by the dotted line. Then, to a first approximation, the nodal points N1 and N2 will move into the positions N' and N,' and the image PQ, will move in the same plane into the position P2'Q' obtained by drawing N,'P,' and Ng'Q2, parallel to PIN' and QIN,' respectively.

Let

Then

N10=1, N,N1=a, N1P1=u, N2P2=v.

N1N,' 10 and N2N2'= (a+1)0.

Hence the displacement of the image is given by

where m 2, the magnification.

и

· · (1)

In order that the displacement of the image may be zero for a given value of 0 we must have

There is thus one, and only one, position of the axis of rotation for which there will be no displacement of the image, viz. that which divides the distance between the nodal points externally in a ratio equal to the value of the magnificationa result obtained in another way by Prof. Anderson. The best position for the axis of rotation will, as already pointed out, be that for which is a maximum subject to the con

ds

αι

dition given by equation (2).

Differentiating (1) and substituting from (2), we get

The rate of change of the displacement thus varies directly as a, the distance between the nodal points and inversely as 1, the distance of the nul point from the first nodal point.

It is greatest when 70, i. e. when mo, and the nul point coincides with N1. It thus appears that the best