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and by operating a second time with ▼

▼2 (rkt)=k(k − 2)rk−4 p2t — 3krk−2t + 2krk¬2Sp\t+r1\t

= pk▼2t − k (2n+ k + 1) μ·k−2t,

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(9)

because p22, and Sp▼t=-nt by Euler's theorem. If we prefer to put V2▲ we may state this identity as

Formula (F)

▲(pt) = pk At + k(2n + k + 1)rk−2t,

where t is homogeneous of degree n.

If t is harmonic we may conveniently write
Formula (G)

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where e denotes the constant k(2n+k+1).

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We are now able to prove the existence of the expansion (E) inductively, by showing that if it exists for all polynomials Fr of degree p it exists for all polynomials of degree p+2. Let m=p+2. Then Fm is of degree p. If the theorem is true for degree p we may write

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but by operating with on both sides of (E)

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(10)

m-6

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where the c's are positive constants by formula (G). Since pm-2 we may take as a possible set of values

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By substituting these values in (E), since F is known, Hm is known. It is therefore evident that the expansion (11) is known by comparison with (10). Hence (E) is known. Now a constant and a linear expression are always harmonic. That is, (E) exists when m=0 and when m=1. Hence it exists when m=2 and when m=3, and so universally.

The same inductive argument shows that the expansion (E) is unique; for if expansion in the form (10) is uniquely possible (11) is uniquely possible. But the expansion is unique for polynomials of degree 0 and 1, hence for all polynomials.

As a simple example let it be required to expand a1 in the form

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where the subscripts denote the degrees of the harmonics, Operating with and determining the numerical coefficients on the right by (F),

A12.14H2+20 H2;

whence

A=24=120Ho,

H=}, H2=4(6x2−2r2), H1=x1 — ‡ r2(6x2 — 2r2) — } r*.

It appears that, in general, each of the H's will be expressed in terms of all the H's of lower subscript.

10. Term by term evaluation of the integral.-Let us now use the elementary theory of the potential function to evaluate the integral

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over a sphere of radius a with centre at the origin. This is the same as finding the potential at a point O' due to a volume-density of f(r). Hn throughout the sphere. Suppose first that O' is outside the sphere. Following Maxwell's notation*, we may write H="Y2, where Y is a surface harmonic. The potential at an external point due to a surface-distribution Y, over a sphere of radius a with centre at the origin is known to be 4πаn+2H2

(2n+1)p2n+1

n

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that is

(2n+1)p2+13 and, since H=Ynan at the surface, a distri

bution f(a)H over the surface will give an external potential 4πa2+2ƒ(a) H2

(2n+1j2n+1

For the external potential due to the solid

sphere we therefore have

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the accents being dropped after the integration, i. e. r is put for T(00'). We can now evaluate the integral propounded

* Elect. and Mag. 3rd Ed. vol. i. Art. 131 a.

470 Operator in Combination with Homogeneous Functions.

in Art. 8 when the point O' is outside the sphere by applying (E) and (H) in succession, viz.

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To illustrate, let us find the external potential when the density of the solid sphere varies as the fourth power of the distance from a diametral plane. By the expansion of a1 already obtained

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where the values of the H's are those calculated in Art. 9 for the expansion of a1, and r is written for T(00′) on the right.

11. The potential inside the sphere.-It is known that a surface-distribution Y, over a sphere of radius a and centre

at the origin causes a potential inside the sphere

or

4π"Y

(2n+1)an-1

4TH2 The potential at a point inside the solid (2n+1)a"-1 sphere may be regarded as due to two additive causes, first a solid sphere of radius r on whose surface the point lies, second a shell of thickness ar fitting outside the first. The first part of the resulting potential is, by (H),

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and the second part, due to a surface-density f(r)H2dr on each infinitesimal shell of radius greater than r, is by the formula quoted at the beginning of this article,

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(2n+1)√, rf(r) dr.

Hence the potential inside the solid sphere due to volumedistribution f(H)m is given by

Formula (A')

SSS = ƒ (1) Hnd V =

4πHn r 1

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(2n + 1) { p2n+1√ " p2a+2ƒ(r) dr + {“ rf(r) dr},

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(H')

whence by means of the expansion (E) we can find the value of (7) when the point is inside the sphere.

Completing the application to a distribution as we have the internal potential

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As a verification, the internal and external expansions coincide if we write ra. The external potential satisfies Laplace's equation while the other yields 4 when operated on with A, agreeing with Poisson's equation*.

LV. On Graphical Methods of correcting Telescopic Objectives. By A. O. ALLEN, Lecturer in Optics, The University of Leeds t.

[AVING had occasion recently to use the N.P.L. tables

relating to small objectives, it occurred to me that the information there furnished, as well as much more of equal or even greater importance, could be given in a very small compass by means of a few formulæ, in combination with graphical methods. It is true that this means substituting calculation for a direct extraction of values from tables, but a number of considerations may be set off against this. First, the calculations I propose are quite simple; most of them are also fairly short. Such as they are, they are not likely to act as a deterrent; for it must be remembered that both the tables and the equivalent calculations lead to figures such as no manufacturer with a reputation to keep up would employ. It may safely be assumed that in future all lens-makers will use the services of an expert computer, and the labour of computing is so great in any case that a little more at the outset will not be objected to, especially if that little extra work saves a great deal of labour further on. Again, the tables are only for a few selected glasses; no tables of reasonable bulk could include all available glasses, and even with these few it is necessary to apply sundry corrections for variations of refractive index and dispersive In general the polynomial

„IIm-4 +...

r2Hm r4Hm-2
+
2(2m+3)+4(2m+1) C(2m—1)

yields the right member of (E) when operated on by A, (proof by (F)). Communicated by the Author.

power. All the lenses are achromatic doublets for C and F, whereas in practice the objective will often be required to have some chromatic error. All the lenses are cemented, so that they are corrected either for sphericity or for coma, but not for both; no tables of moderate bulk could include the possibility of air-gaps.

Again, all the lenses are computed for an object at infinity; whereas reading-telescopes should be computed for a comparatively near object. Finally, the tables refer only to doublets, whereas the methods given below can be applied also to triplets; or, for that matter, to systems with any number of thin components, but combinations of four or five lenses as telescopic objectives are to be regarded as mere scientific bizarreries. Now all these variables (refractive index, dispersive power, position of object, air-gaps, number of components) are taken account of below, and without any serious addition to the labour involved. But it must always be remembered that the results arrived at in this way (or by the tables) ought never to be seen by the lens-grinder; they are simply intended to give the computer a favourable start.

The assumptions made are: (1) that the thickness of each lens or gap is negligible; (2) that all the angles in the calculation are so small that the excess of any angle above its sine is exactly equal to a sixth of the cube of the angle. In other words, the rays could all travel within a capillary tube lying along the axis of the lens.

The symbols employed below are chosen to suit the present problem, and would not necessarily fit into a more general scheme. Focal lengths, radii, and intersection-distances are avoided; it is the reciprocals of these quantities which are more important. The four curvatures are C1, C2, C3, C4, from left to right; the powers of the two lenses are P1, P2, and it is assumed that the power of the combination is chosen as a unit, so that pi+p=1. So far as this paper is concerned there is no condition whatever connecting Pi and p2; they may be quite independent, or may be chosen to give achromatism between any two colours, or to give a desired chromatic error, or to satisfy some other condition not stated; they may be of like or unlike signs. If a ray incident on the system is converging toward a point beyond the system, the reciprocal of its intersection-distance will give the initial convergence, u.

After the ray has passed through the first surface the quantity becomes u; and as the thickness of the lens is neglected, uu, and so on. All these c's, u's and p's are to be thought of as "angles per unit height of incidence." The excess of

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