There is another way of thinking of an ellipse which is also useful; it may be thought of as a circle which has been squashed. Suppose for instance that you took a wooden hoop and stood it up and put a weight on the top of it; the hoop would get squashed into more or less the shape of an ellipse. In the figure, the hoop is drawn circular, as it is before the weight is put on; then a heavy weight is put on the highest point, and the hoop takes more or less the form of the dotted curve in the figure. The weight, which was put on at , has made the top of the hoop sink to . The hoop is supposed to be fastened, like a wheel, on to an axle in the middle, . An ellipse can be obtained from a circle which is standing upright by diminishing all [Pg 76] vertical distances in a certain fixed proportion; that is to say, if is any point of the circle, which is at a height above the level of the axle, we go down to a point below , such that the height bears a fixed ratio, to , the same, of course, as the ratio of to . The ratio of to is also of course the same for any point of the curve, and equal to the ratio of to . We will call this the amount of “flattening” of the ellipse. This is not a recognized expression, but will prove convenient for our purposes. To state the whole thing precisely: Given a circle, imagine it to be stood upright, like a wheel, with an axle through the centre. Then lower each point in the top half of the wheel by a fixed proportion of its height above the level of the axle, and raise each point in the bottom half in the same proportion of the lowering to the final height (or of the raising to the final depth, in the lower half) we will call the amount of “flattening” in the ellipse. That is to say, if is half of (and half of ), the amount of flattening is a half; if is a third of , the amount of flattening is a third; and so on. We can now explain what are the ellipses which are possible for the electron in a hydrogen atom. [Pg 77] In the figure, represents the electron, represents the nucleus, which is in a focus of the ellipse. is the point where the electron is nearest to the nucleus, the point where it is farthest from it, the centre of the ellipse, which is half way between and . There are now two periodic characteristics of the orbit, instead of only one, as in the case of the circle. The first periodic characteristic is, as before, the angle which makes with . The other is , the distance of the electron from the nucleus. This grows continually smaller while the electron is travelling from to , and then continually larger while it is travelling from to . As there are two periodic characteristics of the orbit, the general quantum-theory will give two conditions that the orbit must fulfil, instead of only one. It is impossible to explain the process by which the results are obtained, [Pg 78] but the results themselves are fairly simple.