The criticisms which may be made of the table are sufficiently obvious, but it may be noted that a great many possible combinations would fall about the region 6 to 7μ and are therefore not separately observable; but it may be that their presence contributes to the great depth and breath of the absorption band at this point, which is exceptionally intense for a single "line," even though a fundamental lies within the limits of the band. One other point of interest is concerned with the band at 3.9 μ, which Schaefer explains as being due to the combination vi+vo, and which should. therefore have the same structure as 2v, which falls at 3·4 μ. In the above table it is suggested that it is formed by 2v,-v', but it is also possible that vi+v+ contributes to the production of the band. Such a superposition would

μ

mask the sharpness of the doublet structure associated with V1, and this is in agreement with observation, for throughout the series of carbonates the band at 3.9 is unresolved, only the merest suggestion of a doublet being given by Schaefer's curves (2), in contrast to that at 3-4 μ, which is a very clearly defined doublet.

At the same time a very awkward detail is that of the relative intensities of the bands at 3.9 and 31, that which in the above table is supposed to be due to 2v,-v' being very strong, whereas that due to 2v+v' is comparatively weak. It is not claimed that the suggestions embodied in the above table are in any sense final, because it is impossible to expect exact numerical agreement between calculated and observed values, so long as a correction analogous to that found by Kratzer (6) in the overtones of gaseous molecules is neglected, as has perforce been the case in this paper.

Summary.

The experiments have shown that infra-red rays are partially polarized by transmission through thin plates of calcite cut parallel to the optic axis, and such polarization effects have been used to determine which of the bands observed in the absorption spectrum of calcite really correspond to maxima of absorption, and which are merely spurious and due to interference effects. It is also suggested that the spectrum can be explained without assuming the inactive frequency to be active in combination.

The apparatus used was that described in previous papers (5) My thanks are specially due to Dr. E. K. Rideal for his continued interest and encouragement.

The Laboratory of Physical Chemistry,
The University of Cambridge.

March 1928.

(1) Taylor & Rideal, Phil. Mag. iv. p. 682 (1927).

(2) Schaefer, Bormuth, & Matossi, Zeit. für Phys. xxxix. p. 648 (1927). (3) Kornfeld, Zeit. für Phys. xxvi. p. 205 (1924).

(4) Schaefer & Schubert, Ann. der Phys. 1. p. 283 (1916).

(5) Taylor & Rideal, Proc. Roy. Soc. A, cxv. p. 589 (1925); Rawlins, Taylor, & Rideal, Zeit. für Phys. xxxix. p. 660 (1926).

(6) Kratzer, Zeit. für Phys. iii. p. 289 (1920).

Phil. Mag. S. 7. Vol. 6. No. 34. July 1928.

H

VII. A Method of Calculation suitable in certain Physical Problems. By T. J. I'A. BROMWICH *.

N certain physical problems, there are many examples of pairs of quantities (say x, y) which are connected by a relation of the type

A and n being constants (independent of x, y). Naturally one would usually reduce such a formula as (1) to the logarithmic form

log y = log A+ n logr

for the purposes of numerical work.

(2)

But, during 1918, I was placed in the position of having to make many calculations of this kind; and the conditions were such that x and y could not differ greatly from certain standard values ao and yo. After some experience in the actual work of the calculations, I convinced myself that a more accurate (as well as a more rapid) method was to calculate the value of the difference (y-yo) in terms of (x-x).

Having recently learned that this transformation could be used with great advantage (in a totally different problem), it has seemed worth while to write out a short account of my method, with the hope that it may prove as useful to other calculators as it has been to myself.

Write, for brevity,

(x-xo)/x=t, (y-yo)/yo=v.

Then the equation (1) leads to

Thus, using the binominal theorem, we have, from (5),

v=nt+n(n−1)+....

(6)

For such problems as can be handled conveniently by this method, t is so small that ts may safely be neglected (and in many cases t is also negligible).

* Communicated by the Author.

these values being independent of x, y.

To illustrate the actual working of the method, let us take an idealised example of the actual problem with which I was concerned in 1918: that is, the adjustment

cordite charges. Without giving exact values, the naval 15-inch guns (in 1914-1918) may be represented by taking yo to be a charge of 420 lb. and xo to be a velocity of 2400 f./s. For M.D. cordite the index n may be taken to be -10/7 (with four-figure accuracy).

Thus from (9) we find that

To complete the illustration, let us consider the adjustments necessary for differences of velocity x-x=50 f./s. Then (8) gives on substitution from (10)

Again, if x-x=20 f./s., the second-order terms are negligible and the adjustment is simply 5 lb.

* With large charges the results are only worked out to the nearest lb (4 oz.). The meaning of the last line (for example) is that, if a charge of 420 lb. gives a velocity of 2350 f./s., then 432 lb. 12 oz. should give 2400 f./s. But so large a correction as 50 f./s. would not usually be made without further firing-proofs.

VIII. Some Problems in Electrical Machine Design involving Elliptic Functions. By R. T. COE, M.A., M.Sc.Tech., A.M.I.E.E., and H. W. TAYLOR, M.I.E.E.*

SUMMARY.

N this this paper the authors investigate four problems, machine for various conditions of slotting and grooving on one side of the gap, and the other concerning the temperature distribution in the insulation between two conductors and the sides of a slot. The former have special reference to the subject of pole-face losses, and experimental results available for one of the cases considered are found to be in satisfactory agreement with the theoretical results obtained.

These four problems, which can all be treated as twodimensional, are dealt with by the method of conformal transformation, and, by considering symmetrical figures, the results are obtained in the form of equations involving elliptic functions.

By a systematic treatment the authors show how numerical results may be calculated from such expressions without the difficulties usually associated with elliptic functions.

The results are collected together in the form of curves ready for immediate application to any particular problem.

I. Introduction.

SYNOPSIS.

II. The Variation of Flux-Density on a smooth Pole-face opposite a Succession of Open Slots.

(a) Analysis of Problem.

(b) Practical Deductions.

(i.) Maximum, Minimum, and Mean Flux-densities on smooth Pole-face.

(ii.) The Gap Coefficient K,.

(iii.) Calculation of Flux-density Variation along smooth Pole-face.

Communicated by Sir Richard T. Glazebrook, K.C.B., F R.S.