similarly brief time during which the rate of recession gradually increases from nothing up to the actual rate of the engines' velocities added together. The change of tone may be thus illustrated:*—

A

B

D

A B representing the sound of the approaching whistle, B C representing the rapid degradation of sound as the engine rushes close past the hearer, and C D representing the sound of the receding whistle. Where a bell is sounded on the engine, as in America, the effect is better recognized, as I had repeated occasion to notice during my travels in that country. Probably this is because the tone of a bell is in any case much more clearly recognized than the tone of a railway whistle. The change of tone as a clanging bell is carried swiftly past (by the combined motions of both trains) is not at all of such a nature as to require close attention for its detection.

On

However, the apparent variation of sound produced by rapid approach or recession has been tested by exact experiments. a railway uniting Utrecht and Maarsen "were placed,” the late Professor Nichol wrote, "at intervals of something upwards of a thousand yards, three groups of musicians, who remained motionless during the requisite period. Another musician on the railway sounded at intervals one uniform note; and its effects on the ears of the stationary musicians have been fully published. From these, certainly-from the recorded changes between grave and the more acute, and vice versû―confirming, even numerically, what the relative velocities might have enabled one to predict, it appears justifiable to conclude that the general theory is correct; and that the note of any sound may be greatly modified, if not wholly changed, by the velocity of the individual hearing it," or, he should have added, by the velocity of the source of sound: perhaps more correct than either, is the statement that the note may be altered by the approach or recession of the source of sound, whether that be caused by the motion of the sounding body or of the hearer himself, or of both.

It is difficult, indeed, to understand how doubt can exist in the mind of any one competent to form an opinion on the matter, though, as we shall presently see, some students of science and one or two mathematicians have raised doubts as to the

Even this statement is not mathematically exact. If the rails are straight and parallel, the ratio of approach and recession of an engine on one line, towards or from an engine on the other, is never quite equal to the engines' velocities added together; but the difference amounts practically to nothing, except when the engines are near each other.

validity of the reasoning by which it is shown that a change should occur. That the reasoning is sound cannot, in reality, be questioned, and after careful examination of the arguments urged against it by one or two mathematicians, I can form no other opinion than that these arguments amount really but to an expression of inability to understand the matter. This may seem astonishing, but is explained when we remember that some mathematicians, by devoting their attention too particularly to special departments, lose, to a surprising degree, the power of dealing with subjects (even mathematical ones) outside their department. Apart from the soundness of the reasoning, the facts are unmistakably in accordance with the conclusion to which the reasoning points. Yet some few still entertain doubts, a circumstance which may prove a source of consolation to any who find themselves unable to follow the reasoning on which the effect of approach or recession on wave-lengths depends. Let such remember, however, that experiment in the case of the aerial waves producing sound, accords perfectly with theory, and that the waves which produce light are perfectly analogous (so far as this particular point is concerned) with the waves producing sound.

Ordinary white light, and many kinds of coloured light, may be compared with noise-that is, with a multitude of intermixed sounds. But light of one pure colour may be compared to sound of one determinate note. As the aerial waves producing the effect of one definite tone are all of one length, so the ethereal waves producing light of one definite colour are all of one length. Therefore if we approach or recede from a source of light emitting such waves, effects will result corresponding with what has been described above for the case of waterwaves and sound-waves. If we approach the source of light, or if it approaches us, the waves will be shortened; if we recede from it, or if it recedes from us, the waves will be lengthened. But the colour of light depends on its wave-length precisely as the tone of sound depends on its wave-length. The waves producing red light are longer than those producing orange light, these are longer than the waves producing yellow light; and so the length-waves shorten down from yellow to green, thence to blue, to indigo, and finally to violet. Thus if light shining in reality with a pure green colour, approached the observer with a velocity comparable with that of light, it would seem blue, indigo, or violet according to the rate of approach; whereas if it rapidly receded, it would seem yellow, orange, or red according to the rate of recession.

Unfortunately in one sense, though very fortunately in many much more important respects, the rates of motion among the celestial bodies are not comparable with the velocity of light, but

are always so much less as to be almost rest by comparison. The velocity of light is about 187,000 miles per second, or, according to the measures of the solar system at present in vogue (which will shortly have to give place to somewhat larger measures, the result of observations made upon the recent transit of Venus), about 185,000 miles per second. The swiftest celestial motion of which we have ever had direct evidence was that of the comet of the year 1843, which, at the time of its nearest approach to the sun, was travelling at the rate of about 350 miles per second. This, compared with the velocity of light, is as the motion of a person taking six steps a minute, each less than half a yard long, to the rush of the swiftest express train. No body within our solar system can travel faster than this, the motion of a body falling upon the sun from an infinite distance being only about 370 miles per second when it reaches his surface. And though swifter motions probably exist among the bodies travelling around more massive suns than ours, yet of such motions we can never become cognisant. All the motions taking place among the stars themselves would appear to be very much less in amount. The most swiftly moving sun seems to travel but at the rate of about 50 or 60 miles per second.

Now let us consider how far a motion of 100 miles per second might be expected to modify the colour of pure green light selecting green as the middle colour of the spectrum. The waves producing green light are of such a length, that 47,000 of them scarcely equal in length a single inch. Draw on paper an inch and divide it carefully into ten equal parts, or take such parts from a well-divided rule; divide one of these tenths into ten equal parts, as nearly as the eye will permit you to judge; then one of these parts, or about half the thickness of an average pin, would contain 475 of the waves of pure green light. The same length would equal the length of 440 waves of pure yellow light, and of 511 waves of pure blue light. (The green, yellow, and blue, here spoken of, are understood to be of the precise colour of the middle of the green, yellow, and blue parts of the spectrum.) Thus the green waves must be increased in the proportion of 475 to 440 to give yellow light, or reduced in the proportion of 511 to 475 to give blue light. For the first purpose, the velocity of recession must bear to the velocity of light the proportion which 30 bears to 475, or must be equal to rather more than one sixteenth part of the velocity of lightsay 11,600 miles per second. For the second purpose, the velocity of approach must bear to the velocity of light the proportion which 36 bears to 475, or must be nearly equal to one thirteenth part of the velocity of light-say 14,300 miles per second. But the motions of the stars and other celestial bodies, and also the motions of matter in the sun, and so forth, are very

much less than these. Except in the case of one or two comets (and always dismissing from consideration the amazing apparent velocities with which comets' tails seem to be formed), we may take 100 miles per second as the extreme limit of velocity with which we have to deal, in considering the application of our theory to the motions of recession and approach of celestial bodies. Thus in the case of recession the greatest possible change of colour in pure green light would be equivalent to the difference between the medium green of the spectrum, and the colour 1-116th part of the way from medium green to medium yellow; and in the case of approach, the change would correspond to the difference between the medium green and the colour 1-143rd part of the way from medium green to medium blue. Let any one look at a spectrum of fair length, or even at a correctly tinted painting of the solar spectrum, and note how utterly unrecognizable to ordinary vision is the difference of tint for even the twentieth part of the distance between medium green and medium yellow on one side, or medium blue on the other, and he will recognize how utterly hopeless it would be to attempt to appreciate the change of colour due to the approach or recession of a luminous body shining with pure green light and moving at the tremendous rate of 100 miles per second. It would be hopeless, even though we had the medium green colour and the changed colour, either towards yellow or towards blue, placed side by side for comparison-how much more when the changed colour would have to be compared with the observer's recollection of the medium colour, as seen on some other occasion?

But this is the least important of the difficulties affecting the application of this method by noting change of colour as Doppler originally proposed. Another difficulty which seems somehow to have wholly escaped Doppler's attention, renders the colour test altogether unavailable. We do not get pure light from any of the celestial bodies except certain gaseous clouds or nebulæ. From every sun we get, as from our own sun, all the colours of the rainbow. There may be an excess of some colours and a deficiency of others in any star, so as to give the star a tint, or even a very decided colour. But even a blood-red star, or a deep blue or violet star, does not shine with pure red light, for the spectroscope shows that the star has other colours than those producing the prevailing tint, and it is only the great excess of red rays (all kinds of red, too) or of blue rays (of all kinds), and so on, which makes the star appear red, or blue, and so on, to the eye. By far the greater number of stars or suns show all the colours of the rainbow nearly equally distributed, as in the case of our own sun. Now imagine for a moment a white sun, which had been at rest, to begin suddenly to approach us so rapidly (travelling more than

10,000 miles per second) that the red rays became orange, the orange became yellow, the yellow green, the green blue, the blue indigo, the indigo violet, and that the violet waves became too short to affect the sense of sight. Then, if that were all, that sun, being deprived of the red part of its light, would shine with a slightly bluish tinge, owing to the relative superabundance of rays from the violet end of the spectrum. We should be able to recognize such a change, yet not nearly so distinctly as if that sun had been shining with a pure green light, and suddenly beginning to approach us at the enormous rate just mentioned, changed in colour to full blue. Though, if that sun were all the time approaching us at the enormous rate imagined, we should be quite unable to tell whether its slightly bluish tinge were due to such motion of approach or to some inherent blueness in the light emitted by the star. Similarly, if a white sun suddenly began to recede so rapidly that its violet rays were turned to indigo, indigo to blue, and so on, the orange rays turning to red, and the red rays disappearing altogether, then, if that were all, its light would become slightly reddish, owing to the relative superabundance of light from the red end of the spectrum; and we might distinguish the change, yet not so readily as if a sun shining with pure green light began to recede at the same enormous rate, and so shone with pure yellow light. Though, if that sun were all the time. receding at that enormous rate, we should be quite unable to tell whether its slightly reddish hue were due to such motion of recession or to some inherent redness in its own lustre. But in neither case would that be all. In the former, the red rays would indeed become orange; but the rays beyond the red, which produce no effect upon vision, would be converted into red rays, and fill up the part of the spectrum deserted by the rays originally red. In the latter, the violet rays would indeed become indigo; but the rays beyond the violet, ordinarily producing no effect, would be converted into violet rays, and fill up the part of the spectrum deserted by the rays originally violet. Thus, despite the enormous velocity of approach in one case and of recession in the other, there would be no change whatever in the colour of the sun in either case. All the colours of the rainbow would still be present in the sun's light, and it would therefore still be a white sun.

Doppler's method would thus fail utterly, even though the stars were travelling hither and thither with motions a hundred times greater than the greatest known stellar motions.

This objection to Doppler's theory, as originally proposed, was considered by me in an article on "Coloured Suns" in Fraser's Magazine for January. 1868. His theory, indeed, was originally promulgated not as affording a means of measuring stellar motions, but as a way of accounting for the colours of double stars. It